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Cepheid known first the Aquilae, ,wihrpeet lsia ehi aibe as variables Cepheid classical represents which ), variability the detected Pigott later, years few A ). ooy itnescale distance mology: Keywords. understandi our telescopes. in large improvements extremely further a for discussed outlook be an will with scale distance wavel cosmic re multiple high-precision at on to candles focus standard these will for I scale absolute properties. wi starting pulsational variables and Cepheid evolutionary II type and Lyrae, RR Cepheid, e metallicity calibrations, primary measu distance relations, Period-Luminosity Desp well-defined measurements. ar distance extragalactic stars the and pulsating physics pulsa classical radially These are respectively. variables lations, Lyrae RR and Cepheid Classical for original In Abstract. 2020. Jun 29 accepted 2020; Jun 21 received MS * China. 100871, 1 Bhardwaj Anupam orsodn uhr -al [email protected] E-mail: author. Corresponding al nttt o srnm n srpyis eigUni Peking Astrophysics, and Astronomy for Institute Kavli (0000) β esi(lo)i 72( 1782 in (Algol) Persei mco Ceti Omicron 000 tr:Vrals ehis RLre yeI ehis Sta Cepheids, II Type Lyrae, RR Cepheids, Variables: Stars: #### : 1 δ rMr n o rep- now and Mira or ehi( Cephei ff cs n te ytmtcucranis ee rsn a present I Here, uncertainties. systematic other and ects, Goodricke Goodricke , , VTuiwsfis bevdby observed first was Tauri RV h rttp fTp ICped TC)adi was it and (T2Cs) Cepheids II by discovered Type of prototype the ( by discovered was Herculis BL Ho T2Cs, of resentative ( RLreadTC a efudin found be can T2Cs and Lyrae RR Cepheids, classical of overview historical detailed more yWlemn lmigadrpre in reported and Flemming Wilhelmina by ainbtenterplainpro n luminosity and period pulsation ( their between lation ( Clouds stars. variable of oldest subtypes the well-studied of two therefore represent and stars Lyrae RR and Cepheid the ( but subtypes Lyrae RR Lyrae as separated variables later cluster Bailey the the discovered variables”. from and “cluster GGCs of 1893 the hundreds in in Observatory stars College variable Harvard for search a ated 1 evt Pickering & Leavitt 2015 al. et Pickering 1889 https://www.aavso.org h otsniiepoe o h rcso tla astro- stellar precision the for probes sensitive most the e ff eet ae nteeojcssu objects these on based rements h bevtoso ehisi h Magellanic the in Cepheids of observations The nts h plcto fteecasclplaigstars pulsating classical these of application The engths. meister t hi xesv s ssadr ade hnst their to thanks candles standard as use extensive their ite ogwt bevtoa ytmtc.Iwl summarize will I systematics. observational with long u hsbifitouto eosrtsta the that demonstrates introduction brief this but ) initi- Bailey Solon discovery, this Following ). ettertcladosrainle observational and theoretical cent tefwsdsoee yWlemn Flemming Wilhelmina by discovered was itself go hs lsia ustr nteucmn r of era upcoming the in pulsators classical these of ng om hahsoia nrdcinaddsrbn hi basic their describing and introduction historical a th igsasta rc on n l-g tla popu- stellar old-age and young trace that stars ting est,Y eYa u5 a inDsrc,Beijing District, Dian Hai 5, Lu Yuan He Yi versity, Leavitt 2My2020. May 12 m ( 1929 Sch¨onfeld , , 1901 1908 n h aiblt flong-period of variability the and ) s vlto,Sas siltos Cos- oscillations, Stars: evolution, rs: , 1912 .Hsoial,WVrii was Virginis W Historically, ). e otedsoeyo re- a of discovery the to led ) ( 1866 .Ti eaini commonly is relation This ). tn stars ating .Tesotpro rep- short-period The ). Ceraski ff rfo hi absolute their from er eiwo classical of review aea Smith & Catelan ff rst establish to orts ( 1905 Pickering ) 1 A . RR 1 #### Page 2 of 1 J. Astrophys. Astr. (0000) 000:#### known as “Cepheid Period-Luminosity relation (PLR)” or the Leavitt Law honouring the discoverer. Ever since, classical Cepheids have played a fundamental role in the extragalactic distance measurements. Ed- win Hubble used Cepheid PLR to determine reliable distance to the M31 and discovered that Andromeda, assumed to be a gaseous nebula at that time, is an- other galaxy beyond our Milky Way (Hubble, 1926). Cepheid-based distances to the galaxies as far as the Virgo cluster allowed Hubble to discover a linear cor- relation between the apparent distances to galaxies and their recessional velocities (Hubble, 1929) - the more distant the galaxy, the faster it moves away from us - now known as the Hubble-Lemaˆıtre law, providing the first evidence of the expanding universe. The slope of the velocity over distance is the Hubble constant (H0), which parameterizes the current expansion rate of the Universe. The current H0 values in the late evolution- ary universe are in tension with early universe measure- ments (Riess et al., 2018a; Planck Collaboration et al., 2018) and therefore understanding the systematics in- volved in standard candles is critical to resolve the H 0 Figure 1. Hertzsprung-Russell diagram displaying schematic tension, and improve the precision of cosmic distance representation of classical pulsating variable stars. A mod- scale. On the other hand, RR Lyrae, which are exclu- ified version of figure taken from Jeffery & Saio (2016) sively old and metal-poor stars, have been used as stel- is shown. The line-shaded regions represent approximate lar tracers of the age, metallicity, extinction and struc- location of variables and the color represents approximate ture of our Galaxy but their use as robust distance in- spectral class mentioned on the top. The zero-age main dicators gained importance more recently thanks to the sequence (ZAMS) and the horizontal branch (ZAHB) are boost of near-infrared (NIR) observations over the last shown with solid and dashed red lines. Cepheid instability two decades. strip is shown with vertical black dashed lines. Dotted lines The goal of this review is to focus on recent represent evolutionary tracks of stars with different masses. progress on absolute calibration of classical Cepheids, The label on the left of the ZAMS shows the stellar mass of RR Lyrae and T2Cs, and their application to the each track. extragalactic distance scale. I strongly emphasize here that a short review can not fully describe all the aspects of these classical pulsating stars as standard 2006; Groenewegen & Jurkovic, 2017b; Jurkovic, 2018, and references therein) are not included in this candles. The interested readers are referred to the review. books, for example, Catelan & Smith (2015) on pul- This review is organised as follows: I describe sating variables and de Grijs (2011) on introduction briefly the description of evolutionary and pulsational to the cosmic distance scale. Additionally, several scenario for classical pulsating stars in Section 2 and excellent reviews are also available in the literature their light curve variations in Section 3. The Sections 4 (Madore & Freedman, 1991; Feast, 1999; Wallerstein, to 6 focus on classical Cepheids, RR Lyrae and T2Cs 2002; Sandage & Tammann, 2006; Catelan, 2009; as distance indicators both from the observational and Feast, 2013; Subramanian et al., 2017; Beaton et al., 2018, and references within). McWilliam (2011) theoretical perspectives at multiple wavelengths. The absolute scale for each standard candle and associated published an excellent set of online conference review systematics is also addressed. Finally, summary with articles on RR Lyrae stars focussed on different aspects an outlook for the future will be briefly presented in beyond their use as distance indicators while a recent Section 7. review of Cepheid and RR Lyrae as young and old stellar population tracers of the Galactic structure can be found in Matsunaga et al. (2018) and Kunder et al. 2. Evolutionary and Pulsational Scenario (2018), respectively. Note that while classical and T2Cs will be discussed extensively here, Anomalous Cepheid and RR Lyrae represent radially pulsating Cepheids (see, Wallerstein, 2002; Fiorentino et al., class of variable stars. Classical Cepheids are young J. Astrophys. Astr. (0000)000:#### Page 3 of 1 ####

( 10-300 Myr), intermediate-mass ( 3-10M ), metal- ⊙ 12 rich∼ stars while RR Lyrae are old ( 10∼ Gyr), low-mass ( 0.5-0.8M ) metal-poor stars. T2Cs≥ also belong to LMC CEP ∼ ⊙ RRL old, low-mass, metal-poor stellar populations. Clas- T2C sical pulsating variables populate a well-defined nar- row vertical region in temperature in the Hertzsprung- Russell (HR) diagram, known as the instability strip (IS). Fig. 1 shows the location of classical pulsat- 14 ing stars including Cepheid and RR Lyrae within the IS in the HR diagram. Classical Cepheids, repre- sented by the prototype δ Cep, are luminous yellow giant variables that pulsate in fundamental (FU), first- overtone (FO), second-overtone harmonics and multi- periodic (double/triple) modes (Soszy´nski et al., 2015). RR Lyrae occupy the region between the cross-section I 16 of the Horizontal Branch (HB) and the IS. Although RR Lyrae stars also pulsate primarily in the fundamental- mode (RRab) and first-overtone modes (RRc), few variables pulsating in more than one mode simultane- ously (RRd) have also been discovered (for example, Soszy´nski et al., 2017b). 18 The T2Cs represent different evolutionary states from post HB to the asymptotic giant branch (AGB) phase and a preliminary classification is done based on their pulsation periods: BL Herculis (BL Her, 1 . P . 4 d), W Virginis (W Vir, 4 . P . 20 d) and RV Tauri (RV Tau, P & 20 d). Soszy´nski et al. (2008) sug- 20 gested another subtype, peculiar W Virginis (pW Vir, 4 . P . 10 d), with distinct light curves and these 0 1 2 peculiar stars are mostly brighter and bluer than W V-I Vir. T2Cs primarily pulsate in the fundamental mode but BL Hers pulsating in the first-overtone mode have Figure 2. Optical color-magnitude diagram for the LMC also been discovered by Soszy´nski et al. (2019). Fig. 2 with data from the Magellanic Clouds Photometric Survey shows distribution of classical Cepheids, RR Lyrae, (Zaritsky et al., 2004) without any extinction corrections. and T2Cs on the observed color-magnitude diagram in Classical Cepheids, RR Lyrae and T2Cs are also overplotted the Large Magellanic Cloud (LMC) from the optical using data from the OGLE survey (Soszy´nski et al., 2015, gravitational lensing experiment (OGLE, Udalski et al., 2016, 2017b). Only the central clusters of each of these 1993; Soszy´nski et al., 2015, 2016, 2018). The T2C pulsating stars are shown for visualization purposes. population is located along the IS and have luminosi- ties that are intermediate between classical cepheids and RR Lyrae. However, the RV Tau and some W Vir overlap the region of classical Cepheids but the to become a red giant with a temporarily inert he- T2Cs are typically significantly less abundant than clas- lium core that is surrounded by a hydrogen burn- sical Cepheids and RR Lyrae. The basic properties ing shell. The expansion of a star happens very of Cepheids and RR Lyrae are given in Table 1. De- rapidly, and therefore, it is difficult to observe it dur- pending on the pulsation periods, classicalCepheids are ing this short evolutionary phase which reflects in the systematically 2-3 magnitude brighter than T2Cs at a Hertzsprung gap between main sequence and red gi- fixed period and∼ up to 8 mag brighter than RR Lyrae. ant stars (Kippenhahn & Weigert, 1991). For a classical ∼ Cepheid-like star (say 5M ) the expansion of stellar ∼ ⊙ 2.1 Stellar evolutionary states envelope moves star to cooler temperature in the HR diagram during the first crossing through the IS. The Let us first consider the evolution of Cepheid-like first crossing is usually very rapid (103-104 years) and intermediate-mass ( 3-10M ) stars in their post main- the star exits the red edge of the IS while the hydrogen ∼ ⊙ sequence phase and going through the IS. Once a shell is still burning. Once the ignition of the helium star has exhausted hydrogen in the core, it expands starts, the star contracts and heats up, and makes a loop #### Page 4 of 1 J. Astrophys. Astr. (0000) 000:####

Table 1. Basic Properties of Cepheid and RR Lyrae variables.

Star Subtype Mass Period range Period MV MK ∆I M days days mag mag mag ⊙ Classical Cepheids Pop I Fundamentalmode(FU) 3-10 1-100 1 -1.5 -2.5 0.45 ∼ 10 ∼ -4.0 ∼ -6.0 ∼ 0.20 50 ∼ -6.0 ∼ -8.0 ∼ 0.65 First-overtone mode (FO) 3-10 0.5-6 1 ∼ -1.5 ∼ -3.0 ∼ 0.20 ∼ 5 ∼ -4.0 ∼ -5.5 ∼ 0.20 ∼ ∼ ∼ RRLyrae PopII Fundamentalmode(RRab) 0.5-0.8 0.3-1.0 0.4 +0.8 -0.1 0.80 ∼ 0.6 ∼ +0.8 ∼ -0.5 ∼ 0.35 First-overtone mode (RRc) 0.5-0.8 0.2-0.5 0.3 ∼ +0.7 ∼ -0.1 ∼ 0.25 ∼ ∼ ∼ ∼ TypeIICepheids PopII BLHerculis(BLHer) 0.5-0.6 1-4 1 +0.2 -1.0 0.50 W Virginis (W Vir) ∼ < 1 4-20 10 ∼ -1.3 ∼ -3.5 ∼ 0.25 RV Tauri (RV Tau) ∼< 1 20-80 50 ∼ -4.0 ∼ -5.5 ∼ 0.30 ∼ ∼ ∼ ∼

Notes: The reader should be cautious regarding numbers shown in this table which are only crude approximation and presented here for a relative comparison. Population I Cepheids are young (10-300 Myr) and Population II RR Lyrae are old (& 10 Gyr) stellar populations. The period-range and I-band amplitudes corresponding to the period listed in the column 4 are estimated within 90% percentile range from the OGLE-LMC data (Soszy´nski et al., 2015, 2016, 2018). Absolute V-band and K-band magnitudes for the given period in the column 4 are derived from the LMC PLRs discussed in the next sections. towards the hotter effective temperature in the HR dia- in greater numbers than long-period ones if both are gram. During this phase, the star crosses the IS for the within the observational limits. second time and undergoes a “blue loop” (See Chap- RR Lyrae, similar to classical Cepheids, are core ter 31, Figures 31.2 & 31.4 in Kippenhahn & Weigert, helium burning stars and occupy a region in the HR di- 1991). Since the central helium burning evolutionary agram which is the intersection between the Cepheid phase lasts for a longer time-scale, the star remains in IS and the HB. A low-mass ( 1M ) star evolves to be- the IS for a greater time than the first crossing. The star come a red giant in its post∼ main-sequence⊙ phase and can undergo a third crossing through the IS during the enters the HB evolutionary phase with helium burning blue loop or return without crossing the blue edge of core. The morphology of the HB itself is quite com- the IS. The exact location of the blue loops is a func- plex and a broad spectrum of HB-related topics are cov- tion of stellar mass and of the chemical composition. ered in the review by Catelan (2009). The zero-age HB For higher mass stars, the extent of blue loops increases (ZAHB) star is characterized by the helium-burning in while the low-mass stars can undergo only one crossing the core and the hydrogen shell burning surrounding through the IS (see Fig. 1). At the late stage of evolution the helium core. The location of ZAHB stars on an of high mass stars, the stellar core contains a degenerate almost horizontal locus in the HR diagram for given mixture of carbon and oxygen which can ignite a super- helium core mass and envelope composition depends nova explosion if the mass limit reaches 1.4M . While on the total mass (or the envelope mass). These stars the initial mass of Cepheids for this to happen⊙ is not have a wide range of effective temperatures such that well constrained, typically an intermediate-mass star massive envelopes lead to cooler temperatures. After evolves onto the AGB while the most massive Cepheid the onset of degenerate central helium burning, only can become a supernova. Interested readers are re- stars with initial main-sequence masses of . 0.8M ferred to Kippenhahn & Weigert (1991); Chiosi et al. achieve the temperatures that place them within the IS.⊙ (1992); Bono et al. (2000); Salaris & Cassisi (2005); Such stars pulsate and become RR Lyrae variables ei- Anderson et al. (2014); Catelan & Smith (2015, and ther when they are close to the ZAHB or else when they references therein) for more details regarding the evo- evolve to the blue or red side in the HR diagram. The lution of intermediate-mass star in the central-helium blue edge of the IS of RR Lyrae is located at an effec- burning phase. Similar to the stellar evolutionary tive temperature of 7200K at the ZAHB luminosity timescale, the time spent in a Cepheid phase decreases level which decreases∼ with increasing luminosity. The dramatically as a function of mass. Note that the red edge of the IS is located somewhere around 5900 higher mass stars have longer pulsation periods. There- K and is very sensitive to the efficiency of convection, fore, short-period Cepheid variables are discovered and the topology of the IS is also dependent on the J. Astrophys. Astr. (0000)000:#### Page 5 of 1 #### metal abundance (see details in Bono & Stellingwerf, pulsations in stars came much later when more detailed 1994; Bono, Incerpi & Marconi, 1996; Bono et al., investigations showed that the above relation is also 1997; Salaris & Cassisi, 2005; Catelan & Smith, 2015; valid for real stars. Marconi et al., 2015, 2018). Around early twentieth century, the periodic T2Cs are in a post-HB evolutionary phase of low- changes in the light and velocity curves of δ Cephei mass stars evolving up the AGB. After the exhaustion favoured the explanation that Cepheids were binary of helium in the core, HB stars move towards brighter stars but the light variations of δ Cephei were signif- luminosities in the HR diagram evolving mainly into icantly different from the confirmed spectroscopic bi- AGB. The post-HB evolution of star depends on its nary Algol. Later, Shapley (1914) presented strong ev- location on the HB or on the effective temperature. idence against binary hypothesis noting that small par- T2Cs represent the class of those pulsating stars that allaxes of Cepheids suggest the luminosities and radii evolve from the blue tail of the HB and reach the IS of primary stars are on average 103L and 5R , re- at higher luminosities than those of RR Lyrae. These spectively. These results favoured∼ stellar⊙ pulsation⊙ for stars suffer shell flashes at the boundary between de- causing light variations in Cepheid-like variables. The generate CO core and the helium region. Short-period pulsation hypothesis for a single star was also used by BL Her stars evolve from the HB, bluer than the RR Martin & Plummer (1915) to explain the radial velocity Lyrae gap, to AGB i.e., towards higher luminosity and variations of a RR Lyrae, then known as cluster vari- larger radius in the process of depleting helium in their able. Finally, the most significant progress for the pul- core. The intermediate period W Vir stars begin to un- sating star hypothesis was made by Eddington (1918, dergo helium shell flashes as they reach AGB phase and 1919), who developed a theory of adiabatic oscillations make temporary excursions into the IS (Wallerstein, of a stellar atmosphere. He suggested that every star 2002). However, Groenewegen & Jurkovic (2017a) of intermediate mass will go through a Cepheid phase showed that the evolution of the W Vir subclass is not for a brief time during its life-cycle, and the physics of clear and they may have the binarity origin similar to radial oscillations was presented in Eddington (1926). pW Vir. The long-period RV Tau are thought to rep- Note that a PLR for pulsating stars follows directly resent post-AGB evolution (Wallerstein, 2002). How- from the Stefan-Boltzmann law and the pulsation equa- ever, RV Tau may also evolve from the more mas- tion (2) such that the bolometric magnitudes can be sive and younger objects or represent binary evolu- written as: tion (Groenewegen & Jurkovic, 2017b; Manick et al., 2018). The evolutionary tracks of T2Cs were pioneered Mbol = a + b log P + cTeff , (3) by Gingold (1976) and the updated theoretical calcula- tions were presented by Bono, Caputo & Santolamazza where pulsation period (P) is used assuming its depen- (1997); Bono et al. (2016) and Smolec (2016). dence on stellar mass and radius through equation (1). The observable color term can replace the Teff which 2.2 Stellar pulsation mechanism results in a Period-Luminosity-Color (PLC) relation. In a two-dimensional plane, neglecting color-term, the I will briefly discuss the physical mechanism driving PLRs in a given wavelength (λ) takes the form: the pulsations in Cepheid and RR Lyrae variables. The classical relation between the pulsation period and the Mλ = a + b log P. (4) mean-density of a pulsating gaseous sphere was first developed by Ritter (1879) who demonstrated that for The physical scenarios regarding the main driving a homogeneous sphere experiencing adiabatic radial mechanism behind Cepheid pulsation and stellar struc- pulsation- ture and evolution were explored by various authors (Christy, 1966; Stobie, 1969a; Cox, 1980b). The P (R/g), (1) pulsation occurs in the stellar envelope for a specific ∝ p range of surface effective temperatures i.e., within the where P is the pulsation period, R is the radius and g is IS, a region where stars are unstable to pulsation. For the surface gravity of gaseous sphere. Since, g M/R2 ∝ example, in a Cepheid-like star with temperature near and using relation between mean density (ρ), mass and 6000K, hydrogen ionization zone occurs close to the radius- surface of the star. Further, helium becomes doubly P √ρ = Q, (2) ionized in another zone deeper in the stellar envelope. The increase in the opacity (κ) increases the ionization where Q is the pulsational constant and the equation is in both the hydrogen and helium ionization zones. known as the pulsation equation or the period-mean- Due to cyclic variations in the opacity, the energy density equation. However, the hypothesis of radial is trapped during contraction, favouring instability. #### Page 6 of 1 J. Astrophys. Astr. (0000) 000:####

Since the ionization occurs deep inside the surface multiband light and radial velocity variations. There- of the star, the pressure or excitation beneath drives fore, quantification of light curve structure can allow stellar envelope expansion. The phenomena works a rigorous comparison between observations and the- as a mechanical valve and the expansion reduces the ory and provide constraints for the stellar pulsation opacity and the energy is released. The temperature models (Wood, Arnold & Sebo, 1997; Marconi et al., and pressure drop and the expansion occurs only due 2013b, 2017). Top panels of Fig. 3 show the I-band to momentum of the envelope structure. Finally, star light curves of classical Cepheids, RR Lyrae and T2Cs starts contracting again and the temperature regains its in the LMC from the OGLE survey (Soszy´nski et al., initial value, thus re-starting the pulsation cycle. Since 2015, 2016, 2018). Typical optical light curves of fun- the mechanism responsible for pulsation is mainly the damental mode Cepheids are symmetric with a saw- increase in the opacity of the ionization zones, it is tooth feature while some Cepheids also exhibit “bump” known as the “κ mechanism” (Kippenhahn & Weigert, along their light curves. Hertzsprung (1926) discov- 1991; Salaris & Cassisi, 2005; Catelan & Smith, ered that Galactic Cepheids present a relationship be- 2015). In the case of radial pulsations, if all parts tween the pulsation period and the location of the bump of a star move in and out together, the pulsation along the light curve - known as “Hertzsprung Progres- occurs in fundamental-mode but the star can have an sion”. Classical Cepheids show a bump on the descend- infinite number of modes. Within the IS, classical ing branch of both the light and velocity curves for pe- Cepheid and RR Lyrae variables exhibit pulsations riods between 6 and 16 days and it appears around the during their long-lasting central helium burning phases of maximum light for periods between 9 and evolutionary phase and the pulsations in T2Cs occur 12 days. For longer period Cepheids, the bump fea- during post-HB evolution. As a passing remark, ture appears on the rising branch. The central period of non-radial pulsation and light curve modulations the Hertzsprung progression has been used to constrain have also been discovered in classical pulsating stars models (Bono, Marconi & Stellingwerf, 2000). It de- (for example see, Dziembowski & Mizerski, 2004; pends on the metal-abundance and wavelength such Netzel, Smolec & Moskalik, 2015; Moskalik et al., that it shifts to longer periods with decreasing metallic- 2015; Smolec & Sniegowska´ , 2016; Anderson, 2016, ity or increasing wavelengths (Bhardwaj et al., 2017a). and reference within for more details). Note that “bump Cepheids” are single mode variables It is important to emphasize here that for with strong regularity in their light curves while the so classical Cepheids, evolutionary masses are sys- called “Beat Cepheids’ are mixed-mode variables that tematically larger at the level of 10 20% pulsate in two or more modes simultaneously. than the pulsation masses or masses− derived The shape of fundamental-mode RR Lyrae optical from other independent methods (Cox, 1980a; light curves is more saw-toothed than that of classi- Caputo et al., 2005; Prada Moroni et al., 2012; cal Cepheids. The RRab light curves also exhibit a Neilson, Cantiello & Langer, 2011; Marconi et al., sharp rise from minima to maxima and a distinct bump 2013a, and references therein). This Cepheid mass dis- near the minimum light. The first-overtone Cepheid crepancy originally proposed by Christy (1968); Stobie and RR Lyrae variables display near-sinusoidal varia- (1969a,b) is an open problem. Discovery of classical tions in the light curves even at optical wavelengths. Cepheid in the binary system (Pietrzy´nski et al., 2010) T2Cs generally display complex light curve variations allowed precise dynamical mass estimates which with BL Her showing variations similar to RRab while were found to be consistent with masses derived from W Vir sometimes complement fundamental-mode clas- the pulsation models (see also, Pilecki et al., 2018). sical Cepheids. RV Tau stars exhibit complex light Therefore, non-standard phenomena like mass-loss, curves with varying maxima and minima from cycle- core overshooting and rotation have been explored in to-cycle. At longer wavelengths, both amplitude and evolutionary models for consistency with pulsation phase variations decrease significantly and the skew- masses (Prada Moroni et al., 2012; Anderson et al., ness and acuteness of Cepheid and RR Lyrae light 2014). curves attain a value close to unity implying a nearly symmetric sinusoidal variations as a function of pulsa- tion phase. 3. Light curve morphology Simon & Lee (1981) used Fourier analysis method to study light curve of periodic variables and showed The analysis of the light curve structure of Cepheid that the lower order Fourier coefficients can be used to and RR Lyrae variables is very useful for their iden- describe the structure of Cepheid and RR Lyrae vari- tification and classification. At the same time, pulsa- ables. In brief, a Fourier series can be fitted to the peri- tion models can also be used to successfully predict the J. Astrophys. Astr. (0000)000:#### Page 7 of 1 ####

14.0 12 FU CEP-0027 P = 3.523 d RRab 00487 P = 0.4464 d BL Her 148 P = 2.672 d FO CEP-0006 P = 3.295 d 15.5 RRc 28053 P = 0.3138 d W Vir 208 P = 8.556 d RV Tau 058 P = 2*21.5 d

14.5 14

I-mag 16.5 15.0 16

15.5 17.5 18 0 1 2 0 1 2 0 1 2 Phase Phase Phase

9 0.6 Cep FU BL Her Cep FO W Vir RRab RV Tau RRc 21 21 6 φ R 0.3

0.0 3 -0.5 0.0 0.5 1.0 1.5 2.0 -0.5 0.0 0.5 1.0 1.5 2.0 log(P) [days] log(P) [days]

Figure 3. Top panels: The representative light curves of classical Cepheids, RR Lyrae and T2Cs in the LMC in I-band taken from the OGLE survey (Soszy´nski et al., 2015, 2016, 2018). The OGLE ID, subtype and periods are also listed on the top of each panel. Bottom panles: The I-band Fourier amplitude (R21) and phase parameters (φ21) for classical pulsating stars plotted as a function of logarithm of pulsation period (P). odic light curves in the following form: tively. A comparison of the observed light and velocity curves of classical Cepheids with theoret- ical models was followed in a number of stud- N ies (Simon & Davis, 1983; Simon & Moffett, 1985; m = m0 + Ak sin(2πkx + φk), (5) Stellingwerf & Donohoe, 1986). The phase lag ob- Xk=1 tained from Fourier decomposition of light curves was where, m is the magnitude as a function of the pulsation found to be the most useful parameter for compari- phase (x). The Fourier-fit results in a mean-magnitude son with observations. Later, Jurcsik & Kovacs (1996) derived an empirical relation between period, Fourier (m0) and amplitude (Ak) and phase (φk) coefficients which are used to construct Fourier amplitude ratios phase parameter (φ31), and metallicity for fundamental Ak mode RR Lyrae variables, which is used extensively in and phase differences: Rk1 = ; φk1 = φk iφ1, for k > A1 1 (Bhardwaj et al., 2015). Fourier analysis− of classical deriving photometric metallicities of the statistical sam- Cepheid, RR Lyrae, and T2C light curves were first ples of RR Lyrae with well-sampled light curves (for carried out by Simon & Lee (1981), Simon & Teays example, Pietrukowicz et al., 2015). Fourier analysis of (1982) and Petersen & Diethelm (1986), respec- Cepheid and RR Lyrae have also been used for the clas- #### Page 8 of 1 J. Astrophys. Astr. (0000) 000:#### sification of these variables (for example, Deb & Singh, and RR Lyrae and provided a preliminary estimates of 2009; Kains et al., 2019). The lower-order Fourier pa- physical parameters such as mass, luminosity, temper- rameters contain the most characteristic information ature, radius, and distances to the observed stars in the about the light curve structure and occupy different re- Galaxy and the Magellanic Clouds with a precision lim- gions in period and Fourier parameter planes. ited by a finer grid of models covering entire period In the bottom panel of Fig. 3, Fourier amplitude range. and phase parameters are plotted against the pulsa- At shorter wavelengths, ultravoilet (UV) and tion period. Classical Cepheids display a distinct X-ray studies of classical pulsators are very limited, progression at 10 days in the case of fundamental and aimed at exploring evolutionary, pulsational and mode Cepheids and at 2.5 days in the case of first- atmospheric properties of these variables (for example, overtone mode Cepheids. The sharp changes in the Downes et al., 2004; Engle, 2015; Siegel et al., 2015; Fourier plane at 10 days are attributed to the reso- Neilson et al., 2016; Sachkov, Bertone & Chavez, nance P2/P0 = 0.5, in the normal mode spectrum 2018, and references therein). At UV wavelengths, (Simon & Schmidt, 1976; Simon & Lee, 1981). In case the amplitudes of classical pulsators are significantly of multiwavelength light curves of Cepheids, the phase large (up to 4 mag in RR Lyrae, Kinman & Brown, of maximum-light shifts to later phases as a function 2014; Siegel et al., 2015), which makes their identi- of wavelength (Madore & Freedman, 1991). Similarly, fication and classification easier provided sufficient the Fourier amplitude parameters decrease while the time coverage is available. Combining with the light Fourier phase parameters increase with wavelength at a curves at longer wavelengths, the large amplitudes of given period for both Cepheid and RR Lyrae variables UV light curves can be used to constrain the impact (Bhardwaj et al., 2017a; Das et al., 2018). The Fourier of convective efficiency in the non-linear pulsation parameters of RR Lyrae do not exhibit any significant models. Furthermore, simultaneous model-fits to structure within short-period range, as can be seen in UV, optical and IR data can also provide insight Fig. 3. However, each subclass of T2Cs display a dis- into the physical parameters of these pulsating stars tinct structure on the Fourier parameter plane, and the (Wheatley, Welsh & Browne, 2012). amplitude and phase parameters also overlap with those of classical Cepheids. The modern stellar pulsation models are based on 4. Classical Cepheids as distance indicators nonlinear, radial pulsation codes that account for nonlo- cal and time-dependent treatment of turbulent convec- Over the past century, Cepheid variables have been tion (Stellingwerf, 1982; Bono & Stellingwerf, 1994; used as standard candles with considerable interest in Bono, Marconi & Stellingwerf, 1999). These models determining distances to star-forming galaxies out to accurately predict the observables, including the topol- 40 Mpc. The Cepheid PLRs in the Galaxy and ogy of the IS, pulsation modes, amplitudes, multiband the∼ LMC have played a vital role in calibrating the light and radial velocity variations (Bono et al., 2000; distant type Ia supernovae in the local universe, and Marconi et al., 2013b, 2015). The model-fitting of ob- connecting to the Hubble flow to determine a value served light curves with pulsation models was first car- of H0 (see the review by Freedman & Madore, 2010). ried out by Wood, Arnold & Sebo (1997) resulting in The Hubble Space Telescope (HST) key project on ex- a robust distance to the LMC. Marconi et al. (2013a) tragalactic distance scale utilized traditional Cepheid- performed model-fitting of Cepheids in an eclipsing bi- Supernovae distance ladder to estimate a 10% precise nary system and predicted pulsation masses that are H0 (Freedman et al., 2001), thus, settling a debate on consistent with dynamical estimates, and later extended the factor of two uncertainty in the expansion rate of the model-fitting to multiband light curves of Cepheids universe. In the past decade, Supernovae and H0 for the in the Small Magellanic Cloud (SMC, Marconi et al., Equation of State (SH0ES) project has made a signif- 2017). Bhardwaj et al. (2017a) and Das et al. (2018) icant progress in reducing the systematics in Cepheid- performed a multiwavelength comparison of Cepheid Supernovae distance ladder to 2% (Riess et al., 2011, and RR Lyrae light curve parameters and found that 2016, 2019). However, improved precision of local models are consistent with observations in most period H0 measurements have resulted in a tension with cos- bins. While the theoretical amplitudes are systemati- mic microwave background based Plank mission re- cally larger than the observed amplitudes, this discrep- sults (Planck Collaboration et al., 2018). The current ancy can be remedied by increasing the convective ef- 9% discord in the H0 measurements between two ex- ficiency in the models. Using a machine-learning ap- treme∼ ends of the universe hints at possible new physics proach, Bellinger et al. (2020) compared observed and in the standard model and is one of the key ongoing modelled Fourier light curve parameters of Cepheid problems in modern cosmology (Freedman et al., 2019; J. Astrophys. Astr. (0000)000:#### Page 9 of 1 ####

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14

Magnitudes 14 Magnitudes V V I - 1 I - 1 J - 2 J - 2 H - 3 H - 3 Ks - 4 18 Ks - 4 18 0.5 1.0 1.5 0.0 0.4 0.8 log(P) log(P)

Figure 4. Multiwavelength PLRs of fundamental and first-overtone mode classical Cepheids in the LMC. The optical data is taken from OGLE survey (Soszy´nski et al., 2015) and the near-infrared photometry is adopted from Macri et al. (2015). The dashed/solid lines represent linear regression over entire/long period range only with a break period at 10 days for fundamental-mode Cepheids and at 2.5 days for first-overtone mode Cepheids.

Riess et al., 2019; Verde, Treu & Riess, 2019). Cepheid PLRs is significant ( 0.2 mag in V and 0.15 Although Cepheids have been used successfully for mag in I) due to the finite∼ width of the IS. Once∼ a the cosmic distance scale, their PLRs suffer from sev- color term as a proxy for temperature is included (ex- eral systematic uncertainties that limit achieving a sub- tinction corrections are already applied), the scatter in percent precision in distance determination. The pri- these relations is reduced to within the observational mary source of uncertainty arises from the lack of pre- uncertainties. Optical data for classical Cepheids in cise absolute calibration of Cepheid PLRs in our own the Magellanic Clouds from the OGLE survey have Galaxy. Apart from the statistical photometric uncer- been used to derive PLRs independently in several tainties, metallicity effects on PLRs and extinction cor- studies (Ngeow et al., 2015; Bhardwaj et al., 2016a,c; rections also contribute to the scatter in the Leavitt law Wielg´orski et al., 2017; Gieren et al., 2018). We list thus limiting the precision of distance estimates to indi- the I-band PLRs for fundamental-mode Cepheids in vidual Cepheids. I will now discuss the recent progress the Magellanic Clouds for a relative comparison in the in the Cepheid PLRs and some possible sources of un- form of equation (4): certainties both from theoretical and observational side, wherever possible, in the following sections. I = 16.892 2.997 log(P) (σ = 0.15), LMC − 4.1 Multiband Period-Luminosity Relations I = 17.264 2.947 log(P) (σ = 0.22). (6) SMC − 4.1.1 LMC calibrations: Classical Cepheids in the These relations are adopted from Wielg´orski et al. LMC have played a crucial role in providing the cal- (2017) and the statistical uncertainties on the slopes ibration of the first-rung of the cosmic distance lad- and zero-points are . 0.02 mag. Generally, the der. More than a century after the discovery of zero-point of the PLRs is adopted at 10 days or at Cepheid PLR, Soszy´nski et al. (2017a) claimed to have the mean of underlying period range to minimize the concluded the work by Heneritta Leavitt on identi- correlated errors due the derived slopes. The slopes fying Cepheid variables in the Magellanic Clouds. of I-band Cepheid PLRs in the LMC and SMC are OGLE survey has discovered more than 9500 classical consistent within uncertainties. The optical PLRs of Cepheids in the Magellanic Clouds allowing an empiri- Cepheids have been used extensively for the distance cal derivation of precise PLRs at optical wavelengths. determination (see reviews by Madore & Freedman, Fig. 4 shows PLRs for classical Cepheids at optical 1991; Feast, 1999; Sandage & Tammann, 2006; and NIR wavelengths. The scatter in the optical band Freedman & Madore, 2010). However, significant #### Page 10 of 1 J. Astrophys. Astr. (0000) 000:#### scatter ( 0.2 mag) in the optical PLRs due to the have also been used extensively in deriving PLRs temperature∼ variations, extinction, and metallicity and Cepheid-based distance determinations (for exam- limits their use in the era of precision cosmology. ple, Ita et al., 2004b,a; Inno et al., 2013). Ripepi et al. In the past two decades, significant progress has (2017) also provided time-series NIR photometry for been made in deriving precise PLRs for classical Cepheids in the SMC from the VMC survey. The Cepheids at NIR wavelengths. The pioneering work Ks-band PLRs for Cepheids in the LMC (Macri et al., of McGonegal et al. (1982) showed that the scatter of 2015) and SMC (Ripepi et al., 2017) in the form of Cepheid PLRs even with random phase observations equation (4) are listed below. at NIR wavelengths is almost 2.5 times smaller than at bluer wavelengths. It is well known that tem- K = 16.023 3.247 log(P) (σ = 0.09), perature variations are significantly smaller at longer s LMC − wavelengths and the impact of extinction is about ten Ks SMC = 16.530 3.224 log(P) (σ = 0.17). (7) times less in K-band compared to optical wavelenghts − (Madore & Freedman, 1991). Therefore, both the im- In the Ks-band, the slopes of the PLRs are sim- . pact of differential extinction and the measurement un- ilar within the uncertainties ( 0.02 mag) for LMC certainties on reddening are reduced significantly. Fur- and SMC Cepheids. The scatter in the Ks-band PLRs ther, the pulsation amplitudes are smaller than in the has reduced significantly ( 45% for LMC Cepheids and 25% for SMC Cepheids)∼ as compared to I-band optical bands allowing accurate mean-magnitude de- ∼ termination with sparsely sampled light curves. Also, (equation 6). The difference in the zero-points gives the light curves in the infrared are typically sinusoidal a relative distance between the Clouds and a precise and thus easier to model generating excellent templates. calibration of LMC Cepheid PLRs can be used to esti- This allows for more precise measurement of period mate robust distance to the SMC. At present, the most and mean magnitudes from fewer epochs which is par- precise primary calibration of Cepheid PLRs for dis- ticularly important for more distant systems where deep tance scale studies is based on LMC anchored using observations are very limited. While all these advan- its 1% accurate late-type eclipsing binary distance (µ ∼ = 18.477 0.004 (statistical) 0.026 (system- tages and less sensitivity to metallicity effects makes LMC ± ± infrared PLRs excellent tools for distance determina- atic) mag Pietrzy´nski et al., 2019). tion, smaller amplitude variations also create difficulty in their identification and classification. 4.1.2 Galactic calibrations: Despite the significant One of the earliest statistically significant sample use of Cepheids for extragalactic distance determina- of 92 LMC Cepheids with NIR light curves was pro- tions, the calibrations of Galactic Cepheid PLRs are vided by Persson et al. (2004). The authors also de- not as precise as their LMC counterparts. The main rived Cepheid PLRs and PLC relations with a scat- reason is that the precise geometric distances to Galac- ter of 0.13 mag but their sample predominantly in- tic Cepheids were available only for a small sample cluded∼ long-period Cepheids. The increasingly larger with parallaxes from Hipparcos (van Leeuwen, 2007) sample of Cepheids with NIR time-series are available and HST (Benedict et al., 2007; Riess et al., 2014). with time-domain surveys such as VISTA NIR survey This is changing with increasingly accurate parallaxes of the Magellanic Clouds (VMC, Cioni et al., 2011) from progressive Gaia data releases providing un- precedently precise astrometry (Lindegren et al., 2016; which is targetting almost all OGLE fields in the JKs- bands. Preliminary results on Cepheid PLRs in the Clementini et al., 2017; Ripepi et al., 2018). In the pre- Gaia era, the most accurate parallaxes for Cepheids LMC in JKs-bands from the VMC survey were pro- vided by Ripepi et al. (2012) and Moretti et al. (2014a). were limited to nearby objects (D . 4 kpc with Although, the VMC survey does not cover H-band, it HST, Benedict et al., 2007; Riess et al., 2014, 2018a). is expected to provide near-complete complementary Cepheid distances have also been measured to rel- atively high precision by a number of independent- sample of JKs observations to OGLE Cepheids in the Magellanic Clouds (Ripepi et al., 2017). methods such as the Infrared Surface Brightness tech- Another excellent sample of NIR light curves of nique and Baade-Wesselink methods, cluster main- 1500Cepheidsin the centralbarof the LMC was pro- sequence fitting, and SpectroPhoto-Interferometry (see, vided∼ by the LMC NIR synoptic survey (Macri et al., Gieren, Fouque & Gomez, 1998; Kervella et al., 2004; 2015). Fig. 4 shows the Cepheid PLRs from the survey Fouqu´eet al., 2007; Turner, 2010; Storm et al., 2011; M´erand et al., 2015; Gieren et al., 2018, and references of Macri et al. (2015) where the scatter in Ks band PLR is only 0.08 mag. While this survey provided homo- therein for more details). geneous∼ time-series of Cepheids in the LMC, single- The uncertainties in the available Galactic cali- epoch NIR observations of larger samples of Cepheids brations of Cepheid PLRs are evident from the fact that their application results in a Cepheid-based LMC J. Astrophys. Astr. (0000)000:#### Page 11 of 1 #### distance having systematics typically more than 3% using most empirical calibrations, while a geometric -12 distance to the LMC is now known to 1% precision Z=0.008 WJ,K - 4 (Pietrzy´nski et al., 2019). The different calibrations of Galactic Cepheid PLRs lead to an active debate re- WV,I - 3 garding the universality of Cepheid PLRs between the Galaxy and the LMC as the metallicity and extinction L-2 effects may change the slope as well as the intercept -8 of the PLRs (Sandage & Tammann, 2006). For exam- K-1 ple, a multiwavelength calibration of Galactic Cepheid PLRs was carried out by Fouqu´eet al. (2007) using dis- J tances to Cepheids based on several independent meth-

ods mentioned previously, including trigonometric par- Magnitudes allaxes. The authors did not find any significant varia- -4 I+1 tion in the Cepheid PLRs between the Galaxy and the LMC. Storm et al. (2011) calibrated PLRs using dis- tances derived from infrared surface brightness method and found no variation in the slope and a marginal V+2 change in the zero-point between Galactic and LMC Cepheid PLRs in the NIR bands. Several other stud- 0 ies also provided calibration of Galactic Cepheid PLRs 0.5 1.0 1.5 (Ngeow, 2012; Groenewegen, 2013; Bhardwaj et al., log(P) 2016b) but they all used nearly the same sample of dis- tances to nearby Cepheids. The Galactic Cepheid PLRs based on Baade-Wesselink distances from Gieren et al. Figure 5. Multiwavelength theoretical PLRs of models rep- (2018) differ from their Magellanic Cloud counterparts resentative of fundamental and first-overtone mode Cepheids in the LMC with metal abundance Z=0.008 (Marconi et al., at all wavelengths. The I and Ks-band PLRs from Gieren et al. (2018) are given here in the form of equa- 2013a). Small symbol size represents first-overtone mode tion (4): Cepheids. The dashed lines represent linear regression over the entire period range.

M = 2.149 2.664 log(P) (σ = 0.21), I MW − − M = 2.424 3.258 log(P) (σ = 0.23), (8) Ks MW − − the predicted distance scale. The nonlinear modelling where the uncertainties on the slopes and zero-points of Cepheids incorporating coupling between hydrody- are 0.1 mag and 0.03 mag, respectively. Compar- namical equations and time-dependent convection by ing∼ with the equations∼ (6) and (7), it is evident that the Stellingwerf (1982, 1984); Bono & Stellingwerf (1994) slopes of Ks-band PLRs are similar between the Galaxy formed a solid basis for such comparisons. Bono et al. and the Magellanic Clouds while the slopes of I-band (1999) derived theoretical PLR and PLC relations for PLRs in the Milky Way differs from the ones in the models representative of Cepheids in the Galaxy and Magellanic Clouds but still consistent within 3σ un- the LMC and showed that theoretical VKs-band re- certainty. Bhardwaj et al. (2016b) also provided abso- lations are consistent with empirical investigations. lute calibration of the Galactic relations based on sev- Caputo et al. (2000) extended model computations to eral distance determination methods accounting for the multiple wavelengths and their PLRs were also fairly intrinsic scatter of each technique. The authors de- consistent with observations but also displayed some rived a Ks-band PLR similar to the equation (8) and dependence on metallicity. Bono et al. (2002) also pre- determined an independent distance to the LMC of sented first-overtone Cepheid models in the Magellanic µLMC = 18.47 0.07 (statistical) mag based on NIR Clouds and suggested that a mild overshooting in pul- photometry of Cepheid± from Macri et al. (2015) in con- sation models is needed for the consistency between cordance with the geometric distance. empirical and theoretical PLRs. They did not find any metallicity dependence and estimated distance to the 4.1.3 Theoretical calibrations: Multiwavelength cal- Magellanic Clouds that agree at the 2% level with em- ibrations of Cepheid PLRs based on stellar pulsation pirical results. models have been used to provide comparison with the In stellar pulsation models, for a given chemi- empirical relations and explore possible systematics in cal composition, the major systematics in the abso- #### Page 12 of 1 J. Astrophys. Astr. (0000) 000:####

Galaxy LMC 10 SMC -8 10

12 -6 12 Magnitudes

-4 3.6µm +1 14 3.6µm +1 14 3.6µm +1 4.5µm 4.5µm 4.5µm

0.5 1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5 2.0 log(P) log(P) log(P)

Figure 6. The PLRs of classical Cepheids at mid-infrared wavelengths in the Galaxy, LMC and SMC adopted from Monson et al. (2012), Scowcroft et al. (2011) and Scowcroft et al. (2016a), respectively. The solid lines represent linear regression over entire period range.

lute calibration of Cepheid PLRs arises due to poorly 2014, 2016), and more importantly, can also resolve the understood phenomenon like mass-loss, core over- Cepheid mass discrepancy problem (Stobie, 1969a,b). shooting and rotation. It is very difficult to disen- Fig. 5 displays PLRs at multiple wavelengths tangle the effects of these phenomenon on the mass- for metal-abundance (Z=0.008, Y=0.25) representa- luminosity relation of classical Cepheids adopted as tive of Cepheids in the LMC. The first-overtone input to the pulsation models. Note that the canoni- mode Cepheids are fundamentalized using the equa- cal mass-luminosity relations are those that come from tion: log(PFU ) = log(PFO) + 0.127. The Cepheid stellar evolutionary calculations and the non-canonical models are adopted from Marconi et al. (2013a) and mass-luminosity relations typically have brighter lumi- used in Bhardwaj et al. (2017a). These models include nosity levels by 0.25 dex to account for non-standard Cepheid masses from 4.5 9M adopting both canon- phenomenon (Marconi et al., 2013b). However, the ical and non-canonical mass-luminosity− ⊙ relations, and zero-point of the adopted mass-luminosity also af- both the standard (α = 1.5) and increased convective fects the zero-point of PLRs. For example, an in- efficiency (α = 1.8). The PLRs for fundamental mode crease in the luminosity level by 0.25 dex at fixed Cepheids in the period range, 0.45 < log(P) < 1.45 mass, due to one or more of the above mentioned non- days, are listed below: standard phenomena, implies a decrease of 0.2 mag (10% on distance) in the estimated distance moduli from the PLRs (Marconi, Musella & Fiorentino, 2005; MI = 2.179 2.626 log(P) (σ = 0.19), Theory − − Fiorentino et al., 2007). Furthermore, the zero-point MK = 2.716 3.062 log(P) (σ = 0.11). (9) of theoretical PLRs is also dependent on the treat- s Theory − − ment of convective efficiency through the variation in While the theoretical I & Ks-band Cepheid PLRs the mixing-length parameter in the pulsation models in the LMC are shallower than the empirical calibra- (Fiorentino et al., 2007). I will also discuss the theo- tions in the Magellanic Clouds, I-band PLR is consis- retical predictions of chemical composition on Cepheid tent with the empirical calibration in the Galaxy. Note PLRs later when comparing to empirical investigations. that the slopes of I & Ks-band theoretical PLRs listed Using stellar evolutionary models, Anderson et al. in Bono et al. (Table 2, 2010) are in excellent agree- (2014) investigated the effect of rotation on Cepheids ment with empirical relations but vary significantly be- and found that it affects the mass-luminosity relations tween short (log(P) . 1 day) and long-period (log(P) > particularly during the blue loop phase. The authors 1 day) Cepheids. The slopes of PLRs in the equa- showed that the difference in Cepheid luminosities be- tion (9) are also in agreement with those of long-period tween different crossings of the IS also increases with Cepheids from Bono et al. (2010). Apart from the pe- faster rotation. Furthermore, rotation also contributes riod range under consideration, theoretical PLRs also to the dispersion in Cepheid PLRs (Anderson et al., depend on the composition, adopted mass-luminosity J. Astrophys. Astr. (0000)000:#### Page 13 of 1 #### relation and the efficiency of convection in the pulsa- below: tion models. M = 2.49 3.33 log(P) (σ = 0.09), 4.1.4 Mid-infrared calibrations: The mid-infrared 3.6µm MW − − (MIR) observations of Cepheids hold a significant ad- m = 16.01 3.31 log(P) (σ = 0.11), 3.6µm LMC − vantage with respect to shorter wavelengths because the m3.6µm SMC = 16.50 3.31 log(P) (σ = 0.16). (10) extinction is more than an order of magnitude smaller − (AV 15A3.6µm)at3.6µm band. Furthermore, the lumi- Note that the Galactic calibration was still based on nosity∼ variations due to pulsations are mostly insensi- the HST parallaxes and other independent methods dis- tive to effective temperature. Therefore, amplitude vari- cussed previously but the scatter in MIR Cepheid PLRs ations, which are smaller than K-band, predominantly was reduced to 0.1 mag with a zero-point uncertainty occur from small radius fluctuations. The infrared of only 3%. Equation 10 suggests that 3.6µm-band ∼ Cepheid spectra are also mostly free from line blanket- PLR in the Galaxy and Magellanic Clouds is univer- ing thus reducing the dependenceof the PLRs on metal- sal. The Galactic calibration leads to a precise distance licity, although CO band-head at 4.5µm is very sensitive to the LMC (µLMC = 18.48 0.04 mag) and SMC ± to temperature variations (see Scowcroft et al., 2016b, (µLMC = 18.96 0.04 mag). Scowcroft et al. (2016b) ± for details). Given increasing MIR observations in the also found that ([3.6]-[4.5]) colour is a reliable metal- past decade, several investigations were aimed at pro- licity indicator for Cepheids. The Galactic (zero-point) viding empirical calibrations of MIR PLRs for Cepheid and LMC (slope) calibrations of Cepheid MIR PLRs variables, in particular, with InfraRed Array Camera led to a factor of three decrease in the systematic un- (IRAC, Fazio et al., 2004) onboard Spitzer Space Tele- certainties resulting in a 2.8% precise H0 measurement scope. (Freedman et al., 2012). The absolute calibrations of High-precision MIR photometry for Cepheids in Cepheid PLRs at MIR wavelengths will be critical in the Galaxy and the LMC have been used to derive the era of James Webb Space Telescope (JWST) thanks empirical PLRs at these wavelengths (Freedman et al., to the higher resolution and higher sensitivity enabling 2008; Ngeow, Kanbur & Nanthakumar, 2008; access to crowded and extincted regions of more distant Madore & Freedman, 2009; Marengo et al., 2010). supernovae host galaxies. Most of these studies utilized single-epoch pho- tometry at 3.6, 4.5, 5.8 and 8.0 µm for Cepheids 4.2 Period-Wesenheit relations and the resulting PLRs exhibited a dispersion of 0.15 mag, better than the optical counterparts with Multiwavelength observations of Cepheids (or RR mean-magnitudes∼ from well-sampled light curves. Lyrae) allow us to obtain distances and color excess si- Marengo et al. (2010) used two random epochs of multaneously. Given a reddening law and photometry photometry and provided Cepheid MIR PLRs in- in at least two filters, PLRs can be used to solve for two cluding first-time ever at 24 and 70 µm wavelengths. unknowns - distance modulus (µ) and extinction (Aλ). The zero-points of their Galactic calibrations were Similar to this approach, to circumvent the problem of primarily anchored using the HST parallaxes from extinction, van den Bergh (1975); Madore (1982) con- Benedict et al. (2007). The MIR PLRs of Cepheids structed reddening free Wesenheit magnitudes that are were extended to NGC 6822 (Madore & Freedman, used in deriving Period-Wesenheit relations (PWRs). 2009), IC 1613 Freedman et al. (2009), and for the At given wavelengths, say λ1, λ2, λ3, the Wesenheit OGLE sample of fundamental-mode (Ngeow et al., functions can be written in the following form: 2009, 2015) and first-overtone mode Cepheids in the

Magellanic Clouds (Bhardwaj et al., 2016c). λ3 λ2,λ1 W = mλ R (mλ mλ ), Due to significant advantages of MIR observa- λ2,λ1 3 − λ3 2 − 1 tions, Carnegie Hubble Program was aimed at mea- λ ,λ Aλ3 R 2 1 = , (11) λ3 suring a H0 with a precision of 2% using the ab- " E(mλ2 mλ1 )# solute calibration of Cepheid PLRs∼ at 3.6 and 4.5µm − where m represents the mean magnitude at wave- (Freedman et al., 2011). Time-series observations of λi Galactic and Magellanic Clouds Cepheids spanning length λi and λ1 > λ2. Generally, the super- script λ is dropped from Wλ3 for simplicity when over 24 epochs were obtained as part of this pro- 3 λ2,λ1 gram. Fig. 6 shows MIR Cepheid PLRs in the λ1 = λ3. The total-to-selective absorption ratios Galaxy, LMC and the SMC from Monson et al. (2012), are adopted based on a reddening law (for example, Cardelli, Clayton & Mathis, 1989) assuming a value of Scowcroft et al. (2011) and Scowcroft et al. (2016a). B,V The 3.6µm-band PLRs in these three galaxies are listed RV (Fouqu´eet al., 2007; Inno et al., 2013). The We- senheit relations are a proxy for PLC relations such #### Page 14 of 1 J. Astrophys. Astr. (0000) 000:####

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Magnitudes Magnitudes 14 W WV,I V,I H WH - 1 WV,I - 1 V,I W - 2 WJ,H - 2 J,H W - 3 WJ,K - 3 14 J,Ks 16 s W - 4 WH,Ks - 4 H,Ks

0.5 1.0 1.5 0.0 0.4 0.8 log(P) log(P)

Figure 7. Multiwavelength PWRs of fundamental and first-overtone mode classical Cepheids in the LMC. The optical data is taken from OGLE survey (Soszy´nski et al., 2015) and the near-infrared photometry is adopted from Macri et al. (2015). The dashed/solid lines represent linear regression over entire/long period range only with a break period at 10 days for fundamental-mode Cepheids and at 2.5 days for first-overtone mode Cepheids.

that the effects of the width of the IS are reduced due 2017), respectively. The slopes of the PWRs are to the additional color term. Fig. 7 displays optical consistent within their uncertainties (0.1 mag for the and NIR PWRs for classical Cepheids in the LMC MW and 0.02 mag for the Magellanic Clouds). from Bhardwaj et al. (2016b). The optical PWRs for Several theoretical∼ and empirical studies have em- Cepheids in the Magellanic Clouds from the OGLE sur- ployed different combinations of filters to derive PLRs vey are derived as WV,I = I 1.55(V I), and the em- and subsequently estimate Cepheid-based distances pirical relations are listed as− follows: − (Fiorentino et al., 2007; Bono et al., 2010; Ngeow, 2012; Inno et al., 2013; Bhardwaj et al., 2016b, and ref- erences therein). W = 15.904 3.332 log(P) (σ = 0.083), V,I LMC − It is important to emphasize that SH0ES project W = 16.385 3.330 log(P) (σ = 0.146). (12) utilizes WH Wesenheit magnitudes in deriving PWRs V,I SMC − V,I (see Fig. 7) for H0 determination. The use of three The dispersion in the optical Wesenheit (WV,I) is band PWRs leads to smaller dispersion possibly due significantly smaller when compared to optical LMC to lower correlated systematics in photometry used in Cepheid PLRs in the V and I-bands ( 60% and ∼ ∼ the color term. The total-to-selective absorption ratio 30%, respectively, see Fig. 4). Theoretically, NIR and H (RV,I = 0.41) is small and any possible variations in this optical-NIR PWRs have additional advantage because parameter, due to the choice of adopted reddening law, these relations are independent of metal-abundance and do not lead to large systematics in PWRs. Based on linear over the entire period range (Bono et al., 2010). stellar evolutionary models, Anderson et al. (2016) also The most commonly used NIR PWR is defined as suggested that WH Wesenheit leads to smallest scatter W = K . J K V,I J,Ks s 0 69( s), and these relations in the in the PWRs. Galaxy and the− Magellanic− Clouds are: 4.2.1 Comparison of multiband slopes: The slopes W = 2.63 3.36 log(P) (σ = 0.24), of fundamental-mode Cepheid PLRs as a function of J,Ks MW − − wavelength are shown in Fig. 8. The slopes of LMC W = 15.76 3.28 log(P) (σ = 0.08), J,Ks LMC − Cepheid PLRs are adopted from Bhardwaj et al. (VI, WJ,K SMC = 16.36 3.33 log(P) (σ = 0.16), (13) 2016c), Macri et al. (JHK , 2015), Scowcroft et al. s − s (3.6&4.5µm, 2011) and Madore et al. (5.8&8.0µm, which are adopted from Gieren et al. (MW, 2018), 2009). The Galactic calibrations are adopted from Bhardwaj et al. (LMC, 2016b) and Ripepi et al. (SMC, J. Astrophys. Astr. (0000)000:#### Page 15 of 1 ####

Inno et al. (2013) for the SMC Cepheids. The theoret- V I J H K 3.6 4.5 5.8 8.0 ical calibrations are adopted from Bono et al. (2010). -3.5 The slopes of PWRs are in good agreement among different studies except in the case of WJ,H Wesen- heit. The inconsistency in slopes of PLRs and PWRs may be due to, for example, different sample sizes, -3.0 different photometric systems, the uncertainty on the Galaxy reddening correction, and single-epoch versus time-

Slopes of PLRs LMC domain data in different studies. Regardless, the range -2.5 Theory of slopes of the optical and NIR PWRs is signifi- cantly smaller than the multiwavelength Cepheid PLRs 0 2 4 6 8 suggesting that the PWRs are indeed excellent tools λ µ [ m] for Cepheid-based distance measurements. An ex- ample of application of different Cepheid PLRs and -3.6 PWRs is the Araucaria Project (Pietrzy´nski & Gieren, 2006) that has utilized variable stars as standard can- dles to measure distances to several Local Group galax- -3.3 ies (for example, Pietrzy´nski et al., 2007; Gieren et al., 2013; Zgirski et al., 2017), Sculptor Group galaxies Galaxy (Gieren et al., 2005, 2009), and improve the calibration

Slopes of PWRs LMC of extragalactic distance scale. -3.0 SMC Theory

1 2 3 4 5 λ [µm] 4.3 Systematic uncertainties in the Cepheid-based dis- WH W W W W V,I V,I J,H J,Ks H,Ks tance scale 4.3.1 Photometric mean-magnitudes: The photomet- Figure 8. Top: Multiwavelength slopes of Cepheid PLRs ric uncertainties in individual measurements for as a function of wavelength. Bottom: A comparison of the Cepheid variables contribute to the observed dispersion slopes of Cepheid PWRs. Shaded regions give a crude ap- in the PLRs through the estimates of mean-magnitudes. proximation of the range of most commonly derived slopes Despite the increase in NIR observational facilities in of Cepheid PLRs and PWRs for distance measurements. the past decade, infrared time-series is limited and The bigger symbol size represents larger dispersion in the the light curves are typically sparsely sampled. Since underlying PLRs or PWRs. Cepheids cover a wide period range, optimizing a ca- dence to obtain well-sampled light curves without large phase gaps is difficult when having only a few epochs of measurements. In the case of HST observations of Storm et al. (VI, 2011), Bhardwaj et al. (JHKs, 2016b), Cepheids in the supernovae host galaxies at a distance and Marengo et al. (3.6 8.0µm, 2010). The slopes − of 20-40 Mpc, photometric uncertainties due to blend- of the empirical PLRs for Cepheids in the Galaxy and ing alone can be a few tenths of magnitudes and the LMC from different studies are consistent within un- random phase corrections can also amount to 0.15 certainties. The theoretical calibrations are adopted mag of additional errors (Riess et al., 2016). The∼ pho- from Bono et al. (VIJKs, 2010) and Marengo et al. tometric uncertainties in the nearby galaxies are typi- (3.6 8.0µm, 2010) for metal-abundance (Z=0.02) − cally smaller (. 0.1 mag) on individual measurements. representative of Cepheids in the Galaxy. The theoreti- The templates for Cepheid light curves are use- cal slopes of the VIJKs-band PLRs also agree well with ful to estimate precise mean-magnitudes from sparsely empirical relations but difference in the slopes is rela- sampled light curves. Soszy´nski, Gieren & Pietrzy´nski tively larger at wavelengths longer than Ks-band. How- (2005) provided NIR templates for classical Cepheids ever, Marengo et al. (2010) also found that the slopes of based on a small sample of calibrating Cepheids in the Galactic MIR PLRs calibrated based on the astrometric Galaxy and LMC. The new NIR templates for Cepheids distances are in excellent agreement with the theoreti- were provided by Inno et al. (2015) based on a very cal predictions. large set of 800 Galactic and Magellanic Cloud The bottom panel of Fig. 8 displays the slopes Cepheids. These∼ templates are divided in ten period of PWRs from Inno et al. (2013) and Bhardwaj et al. bins to account for a wide range of Cepheid periods (2016b) for LMC Cepheids, Storm et al. (2011) and and allow mean-magnitude estimates with a precision Bhardwaj et al. (2016a) for Galactic calibrations, and ( 0.02 mag) only limited by the intrinsic accuracy of ∼ #### Page 16 of 1 J. Astrophys. Astr. (0000) 000:#### the templates. One new addition to these templates was authors found negligible contribution to the system- the use of the phase of mean-magnitude along the rising atic uncertainties on the H0 estimates between the lin- branch as an anchor of phase zero-point which allows ear and non-linear model of Cepheid PLRs. However, proper sampling of the light-curves of bump Cepheids. considering that Cepheid PLRs in the supernovae host galaxies presently have a typical dispersion more than 4.3.2 Linear versus non-linear period-luminosity three times the scatter in the calibrator LMC PLRs, relations: The application of Cepheid PLRs to the any possible changes in the slope of PLRs need care- distance scale follows a basic assumption that these re- ful investigation when precise relations become avail- lations are linear over the entire period range. The non- able with JWST and the extremely large ground-based linearity of the PLRs has been a subject of many studies telescopes. in the past decade (Tammann, Sandage & Reindl, The theoretical explanation for the cause of 2003; Sandage, Tammann & Reindl, 2004; possible non-linearities in the Cepheid PLRs is Ngeow et al., 2005; Ngeow & Kanbur, 2006a; not well-understood. Kanbur & Ngeow (2005); Ngeow, Kanbur & Nanthakumar, 2008; Ngeow & Kanbur (2006a); Kanbur et al. (2010) argued Garc´ıa-Varela, Sabogal & Ram´ırez-Tannus, 2013; that the changes in the slope of LMC Cepheid period- Bhardwaj et al., 2016a). The Cepheid PLRs in the color relation (and subsequently PLR) as a function LMC exhibit a change in the slope at 10 days for of pulsation phase contribute to the observed non- fundamental-mode Cepheids and at 2.5 days for linearities. The period-color and amplitude-color rela- first-overtone mode Cepheids at optical wavelengths tions of long-period (> 10 days) classical Cepheids in (Bhardwaj et al., 2016a). The short-period break at the LMC exhibit a nearly flat slope at maximum light 2.5 days has been noted for both fundamental and but a non-zero slope at minimum-light (Bhardwaj et al., first-overtone mode Cepheids in the SMC (Bauer et al., 2014). Kanbur et al. (2010, and references therein) re- 1999; Ngeow et al., 2015; Bhardwaj et al., 2016c). The lated these variations in the period-color relations with break in the PLRs at 10 days has also been observed the interaction of hydrogen ionization front and the stel- for Cepheids in M31 (Kodric et al., 2015, 2018). lar photosphere and the properties of the Saha ioniza- Furthermore, possible non-linearities in Cepheid tion equation, and suggested that the changes in the PLRs have been investigated using a number of period-color relations affect the PLRs through PLC re- independent methods including both parametric and lations. However, the changes in the slope of Cepheid non-parametric statistical tests (Kanbur et al., 2007; PLRs are also strongly correlated with the sharp struc- Garc´ıa-Varela, Sabogal & Ram´ırez-Tannus, 2013; tural changes in the Fourier parameters at the break Bhardwaj et al., 2016a). Bhardwaj et al. (2016a) found periods (Bhardwaj et al., 2016a,c). At the same time, evidence of a break at 10 days in optical Cepheid PLRs metallicity is also expected to play a crucial role as and around 18 days in the NIR PLRs in the LMC. metal-poor Cepheids are brighter than their metal-rich However, the authors did not find any significant bias counterparts at fixed period (Romaniello et al., 2008, between distance estimates using linear and non-linear see next subsection). The observed non-linearity at models of PLRs when combining the LMC sample the long-period end can be an observational bias as in- with Cepheids in the supernovae host galaxies. cluding brightest LMC Cepheids from the OGLE shal- Ngeow & Kanbur (2006b) estimated distances to low survey (Ulaczyk et al., 2013) masks the evidence the type Ia supernovae using calibrated linear and non- of non-linearity in optical Cepheid PLRs at 10 days linear Cepheid PLRs and found marginal difference in (Bhardwaj et al., 2016a). the H0 values and corresponding systematic uncertain- ties. In the traditional distance ladder, only long-period 4.3.3 Metallicity effects: One of the most crucial is- Cepheids in the LMC were used for the calibration sues in the Cepheid distance scale is the dependence of zero-point since distant Cepheids observed in the on metallicity of both the slope and zero-point of the supernovae host galaxies predominantly have periods PLRs and PWRs. The validity of the basic assumption greater than 10 days (for example in the SH0ES project, regarding universality of the Cepheid PLRs in differ- Riess et al., 2011). However, a two-slope model for the ent stellar environment critically depends on negligible calibrated Cepheid PLRs can provide a stronger con- metallicity effects. Theoretical studies by Bono et al. straint on the global slope of the PLR and also reduce (1999); Caputo, Marconi & Musella (2000) based on corresponding systematic uncertainty (Bhardwaj et al., non-linear convective models showed that both the zero 2016a). Riess et al. (2016) included several variants point and the slope of the predicted PLRs are signifi- H of non-linear WV,I PWR in their analysis for the de- cantly dependent on metallicity with the amplitude of termination of the H0 including two-slope model with the metallicity effect decreasing at the longer wave- possible break periods at 10 days or 60 days. The lengths. At a given wavelength, the slope becomes J. Astrophys. Astr. (0000)000:#### Page 17 of 1 #### steeper for lower metal-abundances. These models pre- K-band) such that the more metal-poor Cepheids are dicted that at a fixed period, metal-rich Cepheids should intrinsically fainter than their metal-rich counterparts be fainter than the metal-poor ones (Bono et al., 1999). with similar pulsation periods. Groenewegen (2018) Interestingly, the slope of the optical and NIR PWRs used parallaxes for Galactic Cepheids from Gaia is independent of the metal-content (Fiorentino et al., second data release to investigate period-luminosity- 2007; Bono et al., 2010). However, the metallicity metallicity relations and found no significant metallic- dependence of the zero-point of the PWRs depends ity term. The author argued that the significant parallax on the adopted filters and needs to taken into ac- zero-point offset present in Gaia data leads to system- count. Theoretical models also predict a dependence atic uncertainties of the order of 0.15 mag on the dis- on helium of Cepheid PLRs (Fiorentino et al., 2002; tance scale (see also Riess et al., 2018b). Marconi, Musella & Fiorentino, 2005) which was fur- In more distant supernovae host galaxies, it is im- ther investigated by Carini et al. (2017). The latter possible to measure directly Cepheid metallicity from found negligible effect on PLRs based distance esti- individual stars. Hence, the mean-metallicity of the mates and a systematic uncertainty of up to 7% on host (and target) galaxy is adopted to constrain sys- PWRs based distances. Metallicity and helium vari- tematics due to metallicity effect on the H0 estimates. ations simultaneously affect Cepheid (and RR Lyrae) Riess et al. (2016) found a metallicity dependence ( pulsation properties, light curves and the PLRs thus it 0.24 0.06 mag/dex) similar to Kennicutt et al. (1998∼) is difficult to disentangle the two contributions. which− ± ultimately contributes to 0.5% systematics in Empirically, several independent observations H0 determinations. Even after decades of effort the have suggested a wide range of estimates for the metallicity effects on Cepheid PLRs are not well- metallicity sensitivity on Cepheid distance scale (see understood and even the sign of metallicity sensitiv- Table 1, Romaniello et al., 2008) that vary from -0.9 ity is debated. The precise parallaxes from the fu- mag/dex to negligible dependence on metallicity∼ at ture Gaia data releases for Galactic Cepheids with optical wavelengths. The indirect measurements of high-resolution spectra (for example, Andrievsky et al., the metallicity in external galaxies mostly based on 2002; Lemasle et al., 2013; Genovali et al., 2013, 2014, oxygen nebular abundances of H II regions showed that 2015; Proxauf et al., 2018) and spectroscopic abun- the metal-rich Cepheids are brighter than metal-poor dances for Magellanic Cloud Cepheids (Lemasle et al., ones (Kennicutt et al., 1998; Macri et al., 2006), in- 2017; Mancino et al., 2020) are essential to resolve consistent with the predictions of nonlinear convective metallicity systematics in Galactic and LMC calibra- models (Bono et al., 2010). Several other investiga- tion on the Cepheid distance scale. tions based on empirical Cepheid PLRs also found similar results (Tammann, Sandage & Reindl, 2003; 4.3.4 Other systematic uncertainties: The impact of Sandage, Tammann & Reindl, 2004; Storm et al., extinction on Cepheid-based distance measurements 2004; Groenewegen et al., 2004) or negligible metal- has been mitigated by using either Wesenheit functions licity effects Fouqu´eet al. (2007). Based on direct or PLRs at the infrared wavelengths. However, the measurements of iron abundances for individual choice of adopted reddening law also contributes to the Cepheids, Romaniello et al. (2005, 2008) found that possible systematics due to extinction, specially in the Cepheids become fainter as metallicity increases. They regions with differential reddening where the reddening found significant metallicity effects on V-band PLRs law may not be universal (Nishiyama et al., 2006, 2009; such that metal-rich stars are fainter, a result consistent Nataf et al., 2016). For example, D´ek´any et al. (2015) with theoretical predictions. However, no firm con- identified 35 classical Cepheids in the inner part of the clusion concerning the metallicity dependence on the Galactic disc but Matsunaga et al. (2016) showed that Ks-band PLR has been achieved (Romaniello et al., there is lack of youngpopulation in the inner 2.5 kpc re- 2008; Bono et al., 2010). gion of the Galactic disc except the nuclear stellar disk In the last few years, Wielg´orski et al. (2017) uti- (Matsunaga et al., 2011). Matsunaga et al. (2016) esti- lized precise Cepheid PLRs in the Magellanic Clouds mated a large impact of the reddening correction based and found metallicity effects compatible with zero in on different reddening laws even at NIR wavelengths all bands on PLRs and PWRs. Gieren et al. (2018) leading to an overestimate of distances to Cepheids in employed Baade-Wesselink method to determine dis- D´ek´any et al. (2015) thus locating those in the inner tances to Cepheids in the Galaxy and the Magellanic part of the Galactic disc. Clouds and quantified the strictly differential effect of Cepheids in the wide binaries and in open clusters metallicity on Cepheid PLRs by minimizing systematic can also contribute to a possible bias in distance esti- zero-point uncertainties. The authors found a metallic- mates with additional light contribution to photometric ity dependence in all bands ( 0.23 0.06 mag/dex in measurements of extragalactic Cepheids due to blend- ∼ − ± #### Page 18 of 1 J. Astrophys. Astr. (0000) 000:#### ing and changing spatial resolution along the distance diagrams differ from cluster to cluster. These differ- ladder (Anderson & Riess, 2018). The authors found ences in the Bailey diagrams can be associated with the a negligible effect due to stellar companions and a rel- Oosterhoff type (Oosterhoff, 1939) of the globular clus- atively larger effect due to cluster populations which ter. RRab in Oosterhoff I (OoI) clusters have an average amounts to an overestimate of 0.23% in H0 determina- period of 0.55 days and [Fe/H]& 1.5 dex while RRab tions. in Oosterhoff II (OoII) have average− period of 0.65 days Anderson (2019) investigated the impact of time- and [Fe/H]. 1.5 dex (Oosterhoff, 1939; Smith, 1995; dilation on Cepheid light curves becauseredshift dilates Catelan, 2009−; Catelan & Smith, 2015). the periods of variables in distant supernova-hostgalax- Fig. 9 displays Oosterhoff dichotomy in the GGCs. ies relative to periods of those in the calibrator galaxies. There is a distinct gap between OoI and OoII clusters He estimated a bias of 0.27% in the H0 values and ar- in the periods versus metallicity plot. However, some gued that this effect will become increasingly relevant metal-rich bulge clusters (for example, NGC 6441, for Cepheids in more distant galaxies in the near-future. NGC 6388, [Fe/H] -0.6 dex) have a larger value of mean-period of RRab∼ than OoII clusters. While GGCs display an Oosterhoff gap, globular clusters and dwarf 5. RR Lyrae variables as distance indicators galaxies in the Milky Way satellite systems do not show such dichotomy (see Catelan, 2009, for more details). RR Lyrae, being fainter than classical Cepheids, have Sandage (1958), using equation (2), showedthat the ab- been used less for distance determinations. This is solute magnitude of the horizontal branches differ by changing thanks to larger telescopes used for the time- 0.2 mag in V-band between OoII and OoI clusters, for- domain surveys and increasing use of infrared observa- mer being the brighter cluster. Oosterhoff dichotomy tions. RR Lyrae are population II distance indicators can be explained as the difference in the intrinsic lumi- and provide an independent primary calibration, and nosity for the RR Lyrae in two clusters, with the higher an alternate distance ladder to the traditional Cepheid- metallicity OoI clusters being fainter. Supernovae distance scale. Carnegie-Chicago Hubble The right panel of Fig. 9 shows period-amplitude Program aims to use population II RR Lyraes and the diagram of RRab variables in a OoI and OoII clus- tip of the red giant branch stars, and estimate distances ter respectively. It is evident that RR Lyrae in the to the supernovae host galaxies determining H0 with OoII type cluster have longer periods for a given am- a precision comparable to current Cepheid-based esti- plitude. Note that several RR Lyrae stars display mod- mates (Beaton et al., 2016; Freedman et al., 2019). Re- ulations in their amplitudes and phases from cycle- cently, Freedman et al. (2019) determined the tip of the to-cycle, a phenomenon known as the Blazhko effect red giant branch and supernovae based value of H0 with (Blaˇzko, 1907), but the origin of these effects is still a precision of 2.4% that sits midway the Cepheid-based unexplained despite a number of investigations includ- and Planck measurements. Considering ongoing Hub- ing those with unprecedently high-precision photome- ble tension, it is important to independently test or com- try from Kepler (Jurcsik et al., 2009; Kolenberg et al., plement tip of the red giant branch based distance es- 2010; Szab´oet al., 2010; Buchler & Koll´ath, 2011; timates using independent population II distance indi- Skarka, Prudil & Jurcsik, 2020). The Blazhko effect cators such as RR Lyrae variables. I will discuss ba- in RR Lyrae is one of the main sources of scatter sic properties of RR Lyrae that are relevant for distance in the observational period-amplitude diagram shown scale studies and focus on NIR PLRs as useful tools to in Fig. 9. The RR Lyrae models from Marconi et al. determine robust individual distances in the following (2015); Marconi & Minniti (2018); Das et al. (2018), sections. computed at fixed metal content (Z=0.004) and primor- dial helium contents ranging from Y=0.25 to Y=0.40, 5.1 Period-amplitude diagrams are also shown in Fig. 9. Theoretically, Bailey dia- grams can also be used to constrain the helium content At the beginning of the nineteenth century, Solon Bai- of RR Lyrae stars. The helium-enhancement leads to a ley discovered hundreds of variable stars in the globu- systematic shift in periods which primarily occurs due lar clusters and introduced RR Lyrae variables of a, b, to increased luminosity levels for similar masses (see, and c Bailey types (Bailey, 1902). These types are now Rood, 1973; Sweigart & Catelan, 1998; Marconi et al., typically separated in two classes based on their pulsa- 2018, and references therein). Marconi & Minniti tion mode: RRab (or RR0) are pulsating in the funda- (2018) recently derived helium-abundance (Y=0.245) mental radial mode while RRc (or RR1) are pulsating of RR Lyrae population in the Galactic bulge by com- in the first-overtone radial mode. Bailey constructed paring their minimum period with pulsation models. period-amplitude diagrams (or ‘Bailey’ diagrams) for Bailey diagrams for RR Lyrae have also been con- RR Lyrae in the globular clusters and found that these J. Astrophys. Astr. (0000)000:#### Page 19 of 1 ####

0.8 2 ω cen Z=0.004 - Y, M, L bulge, disk 0.25, 0.57, 1.87 old halo young halo 0.30, 0.58, 1.70 0.7 0.40, 0.59, 1.61 OoII

[days] 1 Amp (V) RRab

P 0.6

OoI M3 - OoI ω cen - OoII 0.5 0 -2.5 -2.0 -1.5 -1.0 -0.5 -0.3 -0.2 -0.1 0.0 [Fe/H] logP

Figure 9. Left: The Oosterhoff dichotomy in the Galactic globular clusters is shown plotting mean period of RR Lyrae against the metallicity using data compiled by Catelan (2009). Right: Period-amplitude diagram for RRab in Oosterhoff I (M3) & II(ω Cen) clusters from the catalog of Clement et al. (2001). For a fixed composition (Z=0.004), RR Lyrae models from Marconi et al. (2015); Marconi & Minniti (2018) with different helium abundance, mass and luminosities are also overplotted.

structed at near UV wavelengths (Siegel et al., 2015). lowing form: The large amplitudes in UV can be useful to constrain the composition effects on RR Lyrae pulsation proper- ties. Further investigations are needed to examine the MV = α + β[Fe/H], (14) dependence of UV pulsation properties on metallicity and Oosterhoff classification. where, the slope (β) and the zero-point (α) have been determined through several cali- 5.2 The visual magnitude-metallicity relation brations in the literature (Fernley et al., 1998; Caputo et al., 2000; Clementini et al., 2003; The Oosterhoff dichotomy was later extended to inves- Bono et al., 2003; Muraveva et al., 2018a, and tigate empirical relations between the location of RR references therein). Several investigations have Lyrae stars in the period-amplitude diagram and both also suggested deviations from the linear form absolute magnitude and [Fe/H]. Sandage (1982) de- of M -[Fe/H] relation (see, Caputo et al., 2000; rived an empirical relation between the period shift of V Bono et al., 2003; Catelan, Pritzl & Smith, 2004; a star with a given amplitude from the mean period- Bono, Caputo & Di Criscienzo, 2007), and also pro- amplitude relation and the metallicity. Later, period- posed a quadratic form of RR Lyrae M -[Fe/H] relation amplitude-[Fe/H] relations were used to determine V (Catelan, Pritzl & Smith, 2004; Sandage & Tammann, metallicities for RRab stars (see, Kinemuchi et al., 2006; Bono, Caputo & Di Criscienzo, 2007; 2006; Kunder & Chaboyer, 2009). However, the corre- Muraveva et al., 2018a). With Gaia second data lation between Bailey diagram and [Fe/H] is debated, release, Muraveva et al. (2018a) suggested that the for example, Bono, Caputo & Di Criscienzo (2007) coefficients of metallicity on luminosity is much higher showed that the Oosterhoff dichotomy plays a key role than previous studies in the literature. Although Gaia in determination of period-amplitude diagram rather parallaxes suffer from systematic zero-point offset than the [Fe/H]. which varies with magnitudes, colors and position in The period-amplitude-[Fe/H] relations suggest a the sky (Muraveva et al., 2018a; Riess et al., 2018b), continuous correlation between period and both the lu- the improvement in the precision of parallaxes is minosity and metallicity for RR Lyrae. An empirical significant. For interested readers, linear and quadratic relation between RR Lyrae V-band absolute magnitude form of M -[Fe/H] relation from Muraveva et al. (M ) and stellar metallicity is usually written in the fol- V V (2018a) are provided here - #### Page 20 of 1 J. Astrophys. Astr. (0000) 000:####

with effective temperature such that redder RR Lyrae are brighter in K-band. This results in an empirical M = 1.17( 0.04) + 0.34( 0.03)[Fe/H], V ± ± period-magnitude relation in K-band. Later, Bono et al. MV = 1.19( 0.06) + 0.39( 0.10)[Fe/H] (2001) derived theoretical K-band Period-Luminosity- ± ± + 0.02( 0.04)[Fe/H]2. (15) Metallicity (PLZ) relation and showed that the uncer- ± tainties on the mass and luminosity also do not effect The coefficient of quadratic metallicity term in the the PLRs significantly at this wavelength. MV -[Fe/H] relation is not significant and the zero- The empirical NIR PLRs of RR Lyrae have points are consistent between both linear and quadratic been a subject of several investigations, particu- versions. While this empirical relation is very simple larly in the globular clusters and the Magellanic and useful tool to determine distances, several sources Clouds (Nemec, Nemec & Lutz, 1994; Butler, 2003; of uncertainties affect the precision of distance mea- Dall’Ora et al., 2004; Sollima, Cacciari & Valenti, surements based on this method. Firstly, the redden- 2006; Borissova et al., 2009; Coppola et al., 2011; ing effects are significant at optical wavelengths due Braga et al., 2015; Muraveva et al., 2015, 2018b, to a large total-to-selective absorption ratio in V-band, and references therein). Similar to Cepheids, NIR RV = 3.1 (Cardelli, Clayton & Mathis, 1989). Even observations of RR Lyrae in most of these studies in moderately extincted regions, the effect of redden- are limited to few epochs or sparsely sampled light ing on optical luminosities is typically larger than the curves. Therefore, NIR templates for RR Lyrae are metallicity effects. In regions with heavy and differ- crucial to determine accurate mean-magnitudes and ential extinction, reddening effects are a major draw- derive precise PLRs. Using well-sampled Ks-band back in using MV -[Fe/H] relation for distance diagnos- light curves of RR Lyrae from the VISTA VVV tics. Another important concern is the evolutionary ef- survey (Minniti et al., 2010), Hajdu et al. (2018) used fects on RR Lyrae population. Typically the evolved- principal component analysis to generate J and H-band RR Lyrae have higher luminosities than those of ZAHB mean-magnitudes from single-epoch measurements. RR Lyrae for a given metallicity. However, there is sig- Recently, Braga et al. (2019) derived NIR templates nificant overlap in the color-space for RR Lyrae evolv- of RR Lyrae and showed that 2% precise mean- ing off the ZAHB and the stars on the ZAHB. This magnitudes can be estimated even from single-epoch evolutionary effect results in the broadening of the HB NIR observations. and the distribution of optical magnitudes (Bono et al., Figure 10 shows empirical PLRs in ω Centauri and 1995). Further, systematic uncertainties in the metallic- theoretical PLRs for RR Lyrae variables at multiple ity measurements due to different metallicity scales and wavelengths. The ω Cen is a well-studied cluster in methodologies add another source of uncertainty in dis- terms of RR Lyrae populations (Navarrete et al., 2015; tance measurements with the visual magnitude metal- Braga et al., 2016, 2018) which exhibits a spread in licity relation. Note that the extinction and metallicity metallicity. Regardless of the metallicity contribution, effects lead to large scatter in the PLRs at wavelengths the infrared PLRs do not exhibit large intrinsic dis- shorter than V-band if the dependence on period is sig- persion. The apparent magnitudes in V-band for RR nificant. For example, Siegel et al. (2015) found a sig- Lyrae luminosities are nearly constant as a function of nificant dependence of NUV PLRs on metallicity with period. Since the bolometric correction sensitivity to difference of up to half a magnitude between coolest effective temperature starts playing a role R-band on- RRab stars in M3 and M15 clusters. wards, a true PLR is observed at longer wavelengths (Catelan, Pritzl & Smith, 2004). The empirical PLRs 5.3 Multiband Period-Luminosity relations for the global sample of RRab and RRc variables in IJHKs-bands from Braga et al. (2016) and Braga et al. RR Lyrae are known to exhibit a very tight PLR (2018) are presented: at infrared wavelengths which makes them excel- lent standard candles. Longmore, Fernley & Jameson (1986) were the first to derive an empirical RR Lyrae Iω Cen = 13.56 1.34 log(P) (σ = 0.06), PLR in the K-band. The pulsation equation im- − J = 13.02 1.88 log(P) (σ = 0.04), plies a Period-Luminosity-color relation for RR Lyrae ω Cen − H = 12.69 2.22 log(P) (σ = 0.04), but the use of such relation suffers from uncertain- ω Cen − ties due to evolutionary effects, effective tempera- Ks ω Cen = 12.63 2.38 log(P) (σ = 0.05), (16) ture and metallicity predominantly at optical wave- − lengths. Longmore, Fernley & Jameson (1986) showed where the uncertainties in the slopes and zero-points that a PLR in K-band comes naturally from pulsa- are . 0.02 and . 0.05 mag, respectively. For a refer- tion equation because bolometric corrections increase ence zero-point, a distance modulus to ω Cen is 13.67 ± J. Astrophys. Astr. (0000)000:#### Page 21 of 1 ####

12 -2 ω cen K Z=0.004 L H K J J I I 14 0 V

Magnitudes Magnitudes V

B B

16 2 -0.4 -0.2 0.0 -0.4 -0.2 0.0 log(P) log(P)

Figure 10. Left: Multiwavelength PLRs of RR Lyrae in ω Cen cluster using data from Braga et al. (2016) and Braga et al. (2018). Right: Multiband theoretical PLRs for RR Lyrae using models adopted from Marconi et al. (2015). The solid lines represent linear regression over entire period range. The periods of RRc stars are fundamentalize to include those in the PLR fits. The magnitudes in different bands are offset by some arbitrary amount for visualization purposes and do not exactly correspond to the scale on the y-axis.

0.04 mag (Braga et al., 2018). The spectroscopic Klein et al. (2014) and the latter found the dispersion in metallicities for RR Lyrae in ω Cen (Sollima et al., these relations to be . 0.05 mag. Using theoretical ap- 2006) were used by Braga et al. (2018) to investigate proach, Neeley et al. (2017) used Spitzer observations metallicity dependence on NIR PLRs. The authors of RR Lyrae in M4 to derive PLRs with a dispersion found a correlation between PLR residuals with [Fe/H]. of 0.05 mag in 3.6µm and 4.5µm bands. Recently, In GGCs with marginal metallicity spread, the disper- Muraveva∼ et al. (2018a) also used Spitzer data in LMC sion in NIR PLRs of RR Lyrae is typically 0.05 mag old cluster Reticulum to derive PLRs and estimate a implying an uncertainty of 2.5% in individual∼ distance distance to the LMC as part of the Carnegie RR Lyrae determination. The right panel of Figure 10 displays Program. theoretical PLRs based on RR Lyrae pulsation models of Marconi et al. (2015) for a fixed metal-abundance. It 5.3.1 Metallicity effects: Although extinction and is evident that the V-band absolute magnitude is nearly metallicity effects are expected to be smaller at longer constant as a function of period for different mass- wavelengths, the impact of metal and helium abun- luminosity levels. The theoretical and empirical inves- dance on RR Lyrae PLRs is actively debated. The- tigations on metallicity effects on RR Lyrae PLRs will oretically, Bono et al. (2001) found that the depen- be discussed in the next subsection. dence on the metallicity is quantitatively smaller ( 0.17 Mid-infrared observations of RR Lyrae, similar to mag/dex) in K-band than that in the optical∼ bands classical Cepheids, have indisputable advantages as (> 0.2 mag/dex). Catelan, Pritzl & Smith (2004) de- discussed previously. Klein et al. (2011) utilized Wide rived metal-dependent PLRs for RR Lyrae based on Field Infrared Survey Explorer (WISE) catalog of RR the calculations of synthetic horizontal branch mod- Lyrae to derive PLRs at MIR wavelengths. The cali- els and found a significant metallicity term (0.21-0.17 brations of RR Lyrae PLRs in MIR bands using WISE mag/dex) in IJHKs-bands. Using a new theoretical data were further improved by Madore et al. (2013); framework of RR Lyrae, Marconi et al. (2015) gener- #### Page 22 of 1 J. Astrophys. Astr. (0000) 000:#### ated pulsation models covering a broad range of metal- 5.3.2 Period-Wesenheit relations: The PWRs for RR abundance (Z=0.02 to 0.0001) and derived PLRs. They Lyrae have also been used for distance determina- found a metallicity dependence ( 0. 18 mag/dex) sim- tions to negate the issues related with reddening cor- ilar to Bono et al. (2001), on RR∼ Lyrae NIR PLRs. The rections. Marconi et al. (2015) presented new optical PLZ relations for RR Lyrae are written in the following and NIR PWRs adopting Cardelli, Clayton & Mathis form: (1989) reddening law and a total-to-selective absorp- tion ratio, RV = 3.06. The dual-band PWRs from Marconi et al. (2015) are listed in the form of equation M = α + β log(P) + γ[Fe/H]. (17) (17):

For a relative comparison, IJK-band PLZ relations are provided in the form of equation (17): M 3.06(M M ) = 1.07 2.49 log(P) V T H − B − V T H − − + 0.01[Fe/H] (σ = 0.08),

MK T H 0.69(MJ MK) T H = 1.05 2.50 log(P) MI = 0.07 1.53 log(P) + 0.17[Fe/H] (σ = 0.09), − − − − TH − − + 0.18[Fe/H] (σ = 0.04), M = 0.50 1.90 log(P) + 0.18[Fe/H] (σ = 0.06), JTH − − (19) M = 0.82 2.25 log(P) + 0.18[Fe/H] (σ = 0.04), KTH − − (18) where the NIR Wesenheits display metallicity depen- dence similar to PLRs. Interestingly, optical Wesenheit where TH represents theory and the uncertainties in the using combination of B and V-band is nearly metal- coefficients of PLZ relations are negligible. While the licity independent and this is not true for other com- theoretical studies consistently predict an appreciable bination of filters used to construct Wesenheit func- metallicity dependence on RR Lyrae PLRs, empirical tions. Using this Wesenheit function, it is possible investigations have most often resulted in a marginal to estimate precise distance independent of the un- dependence on metallicity. certainties on the metallicity measurements (see equa- Sollima, Cacciari & Valenti (2006) used RR Lyrae tion (19) and Braga et al., 2016, for an application to in several GGCs to constrain the metallicity depen- ω Cen). However, the dispersion in this optical We- dence and quantified a relatively small dependence senheit function is twice as large compared to NIR of 0.08 mag/dex on [Fe/H]. In the case of LMC PLRs and PWRs. Marconi et al. (2015) derived several RR Lyrae, Borissova et al. (2009) also found a very combinations of triple-band PWRs relations, similar to mild dependence on metallicity in Ks-band by com- H WV,I function used in SH0ES project, but those require bining NIR photometry and spectroscopic metallici- mean-magnitudes in three independent filters. ties for a homogeneous sample of 50 RR stars in As a passing remark, the theoretically predicted the inner regions. Similarly, Muraveva et al. (2015) first-overtone blue edge (FOBE) on the MV log(P) utilized low-dispersion spectroscopic metallicities of plane is also a useful distance indicator for− stellar 70 RRLs in the bar of the LMC with NIR photom- systems that host a statistically significant number of etry from VISTA VMC survey (Cioni et al., 2011) RRc stars (Caputo, 1997). The FOBE is indepen- and found a marginal dependence ( 0.03 0.07 ∼ ± dent of the metallicity. If the blue part of the IS is mag/dex) on [Fe/H] in RR Lyrae PLRs. More re- well-populated and the metallicity is known, assuming cently, Neeley et al. (2019) used Gaia parallaxes for the mass, a period-luminosity-metallicity relation can Galactic field RR Lyrae to derive multiband PLZ re- be derived for the evolutionary FOBE pulsators (see, lations and found that the dispersion in these rela- Caputo, 1997; Caputo et al., 2000; Bono et al., 2003; tions is dominated by the uncertainties in the parallaxes Beaton et al., 2018, for details). despite reproducing the metallicity dependence pre- dicted from models. The high-resolution spectroscopy 5.3.3 Absolute calibrations: The lack of accurate of RR Lyrae has been obtained mostly for the field parallax measurements for RR Lyrae limits the pre- (Clementini et al., 1995; For, Sneden & Preston, 2011; cision of the absolute calibration of PLRs at infrared Nemec et al., 2013; Pancino et al., 2015, and references wavelengths. Unlike classical Cepheids, the calibra- therein) and globular cluster variables (for example, tion based on the LMC exhibits large dispersion due Sollima et al., 2006; Magurno et al., 2018, 2019). In to spread in metallicity distribution of RR Lyrae and its the case of Magellanic Clouds, low-resolution spec- effect on the PLRs. Feast et al. (2008) utilized Hippar- troscopy has been limited to small samples of RR Lyrae cos and HST parallaxes of RR Lyrae itself to provide a stars (e.g. Gratton et al., 2004; Borissova et al., 2004, zero-point calibration. Benedict et al. (2011) presented 2006; Haschke et al., 2012). HST parallaxes for 5 RR Lyrae variables and provided J. Astrophys. Astr. (0000)000:#### Page 23 of 1 #### absolute calibrations in K -band for a PLZ relation. Re- s 40 cently, Muraveva et al. (2018a) provided absolute mag- Bulge nitudes for RR Lyrae in several bands using Gaia as- LMC trometry for 400 stars but also noted a significant zero-point off∼set in the Gaia parallaxes.

The calibration of MIR PLRs of RR N 20 Lyrae in the GGCs were first provided by Dambis, Rastorguev & Zabolotskikh (2014) using WISE data. The authors found two significantly dif- ferent estimates for the zero-points based on statistical 0 and HST trigonometric parallaxes. Neeley et al. (2019) calibrated multiband PLZ relations of RR Lyrae using 1 2 4 10 20 50 photometry obtained from the Carnegie RR Lyrae Period [days] Program and parallaxes from the Gaia second data release for a sample of 55 Galactic field RR Lyrae 1.2 Bulge LMC stars. They found that the scatter in the PLZ relations is significantly large ( 0.2 mag) when compared to theoretical predictions,∼ and is still dominated by uncertainties in the parallaxes from current Gaia data. 0.6

Despite the metallicity uncertainties, NIR PLRs Amp (I) of RR Lyrae have been used extensively to de- termine distances to several stellar systems, for example, GGCs M92 (Del Principe et al., 2005), 0.0 M5 (Coppola et al., 2011), ω cen (Navarrete et al., 0 1 2 2015; Braga et al., 2018), M4 (Braga et al., 2015), log(P) [days] Galactic center (D´ek´any et al., 2013), LMC old cluster Reticulum (Dall’Ora et al., 2004), Magel- lanic Clouds (Ripepi et al., 2012; Moretti et al., Figure 11. Top panel: The period-distribution of T2Cs in the Galactic bulge and the LMC. Bottom panel: The I-band 2014b; Muraveva et al., 2018a), Carina Dwarf amplitude as a function of period for T2Cs in the Galactic (Karczmarek et al., 2015), Fornax (Karczmarek et al., bulge and the LMC. 2017), and IC 1613 (Hatt et al., 2017). While the application of RR Lyrae PLRs and PWRs to measure distances to individual system is not discussed here, the interested readers are referred to the above mentioned two classes of Cepheids eventually resolved a major papers. issue in the H0 determination at that time, which led to the reduction of the spatial and temporal scales of the universe by a factor of two (Baade, 1956). While 6. Type II Cepheids as distance indicators the T2Cs have not been used extensively as distance indicators being fainter than classical Cepheids, they The discovery of T2Cs played a critical role in the re- have played crucial roles as excellent tracers of stellar vision of the extragalactic distance scale. In his pio- evolution and Galactic structure. I will discuss some neering work, Baade (1944) showed that stellar pop- empirical properties of T2Cs and focus on recent up- ulations of the galaxies are either similar to those in dates in their PLRs for distance measurements. There the solar neighborhood (the slow-moving stars i.e. disk are several excellent reviews on T2Cs (Harris, 1985; stars) or those in the globular clusters. This eventually Wallerstein, 2002; Sandage & Tammann, 2006; Welch, led to the classification of Population I and Population 2012; Feast, 2010, 2013; Beaton et al., 2018) that are II stars. In his seminal papers, Baade (1958a,b,c) in- recommended to the interested readers. troduced a difference in the PLRs of the population II Cepheids in the globular clusters and the classical or 6.1 Pulsation properties of T2Cs population I Cepheids in the spiral arms of the galax- ies. The former are T2Cs that represent old, low-mass Similar to the evolutionary status, the pulsation prop- stellar populations. Before their discovery, both young erties of T2Cs are distinct among the three subclasses. and old Cepheid populations had been used in the PLRs The classification into BL Her, W Vir and RV Tau is and distance scale. The distinction of the PLRs for predominantly based on the pulsation period. How- #### Page 24 of 1 J. Astrophys. Astr. (0000) 000:#### ever, the period range for each group is not universal microlensing survey data to discover T2Cs in the and depends on the stellar environment. The top panel Magellanic Clouds and determined PLC relations for of Fig. 11 shows the period distribution for T2Cs in the W Vir and RV Tau variables. In the past decade, OGLE Milky Way bulge and the Magellanic Clouds. It is ev- survey has discovered several Galactic and Magellanic ident that the minima in the period distributions vary Clouds T2Cs and derived solid, optical-band PLRs between bulge and LMC due to their significantly dif- (Soszy´nski et al., 2008, 2010, 2011, 2017b). Sim- ferent metallicities. Soszy´nski et al. (2011) found that ilar to classical Cepheids and RR Lyrae, increased bulge T2Cs are dominated by short-period BL Her stars availability of NIR observations have allowed several which are more luminous than their counterparts in the investigations on T2C PLRs at these wavelengths, Magellanic Clouds. The bottom panel of Fig. 11 dis- where less sensitivity to metallicity and extinc- plays the period-amplitude diagram for T2Cs in dif- tion leads to tighter PLRs (Matsunaga et al., 2006; ferent stellar environments and exhibits different struc- Feast et al., 2008; Groenewegen, Udalski & Bono, tures for each subclass. The amplitudes of BL Her ex- 2008; Ciechanowska et al., 2010; Ripepi et al., 2015; hibit large scatter at a given period similar to RR Lyrae Bhardwaj et al., 2017a, and references within). variables. A sharp rise in the amplitudes for W Vir stars At NIR wavelengths, Matsunaga et al. with 0.8 < log(P) < 1.3 days can be seen while the am- (2006); Matsunaga, Feast & Menzies (2009); plitudes decrease quickly as a function of period for RV Matsunaga, Feast & Soszy´nski (2011) derived NIR Tau stars. The light curves of T2Cs are also quite differ- PLRs for T2Cs in the GGCs and the Magellanic ent from classical Cepheids or RR Lyrae as they exhibit Clouds. The authors found non-universal slopes of complex variations depending on the subclasses. the PLRs in different systems and also noted varying Unlike classical Cepheids and RR Lyrae, theo- frequency of each subtype. Matsunaga et al. (2006) retical studies on T2C pulsation properties are very derived PLRs for T2Cs in the GGCs and found a limited. Earlier studies were limited to linear models linear relation over entire period range with a typical (Wallerstein & Cox, 1984) and non-linear pulsation dispersion of 0.15 mag in JHKs bands. They ob- models without accounting for the convective trans- tained distances to individual GGCs using MV -[Fe/H] port (Fadeev & Fokin, 1985). Full time-dependent relation for horizontal branch stars and showed a convective pulsation models of BL Her stars were consistency between RR Lyrae and T2C distance provided by Bono, Caputo & Santolamazza (1997); scale. Groenewegen, Udalski & Bono (2008) utilized Marconi & Di Criscienzo (2007); Di Criscienzo et al. NIR photometry of T2Cs in the Galactic bulge to (2007). Bono, Caputo & Santolamazza (1997) estimate a distance to the Galactic center. In the showed that T2Cs pulsate primarily in the fun- LMC, Ripepi et al. (2015) derived T2C PLRs in JKs damental mode and their masses decrease with using data from VISTA VMC survey with intrinsic increasing period, and also derived metallicity in- dispersion of 0.13 mag in J and 0.09 mag in K-band. dependent period-luminosity-amplitude relations. More recently, Bhardwaj et al. (2017b) used data from Marconi & Di Criscienzo (2007) presented the topol- the LMC NIR synoptic survey to derive PLRs in ogy of the IS, and light and radial velocity curves for JHKs-bands. Combining with literature data, they BL Her stars. They showed that the first-overtone IS is presented the largest sample to date of T2Cs with very narrow and therefore most T2Cs are fundamental observations and used it to derive PLRs as well as pulsators which was also seen empirically for T2Cs absolute calibration with the known late-type eclipsing in the Magellanic Clouds (Soszynski et al., 2008; binary distance to the LMC. Furthermore, distance Soszy´nski et al., 2018). Similar to classical Cepheids estimates to several GGCs from Matsunaga et al. and RR Lyrae, T2Cs also follow a PLR that can be (2006) based on the horizontal branch morphology derived from the pulsation equation (see, Section 3.3 are within 1σ of the distances obtained by applying of Matsunaga et al., 2006; Di Criscienzo et al., 2007). the LMC calibrations of T2Cs PLRs (Bhardwaj et al., 2017c). 6.2 Period-Luminosity and Period-Wesenheit relations Figure 12 shows optical PWR and IJHK-band PLRs for T2Cs in the LMC. It can be seen that the The optical studies of T2Cs in the GGCs provided evi- PLRs are not linear throughout the period range i.e. dence of a PLRs (Harris, 1985; McNamara, 1995) but for all subclasses as an ensemble. A linear regression their investigations as useful distance indicators peaked over entire period range results in a large dispersion of with modern data from large photometric surveys 0.6&0.4 mag in V and I-band PLRs for T2Cs in the (Nemec, Nemec & Lutz, 1994; Alcock et al., 1998; ∼LMC, which is not useful for precision distance mea- Kubiak & Udalski, 2003; Majaess, Turner & Lane, surements. Even after excluding RV Tau that are dis- 2009; Schmidt et al., 2009, and references therein). tinctly brighter than the BL Her and W Vir stars, result- For example, Alcock et al. (1998) used MACHO J. Astrophys. Astr. (0000)000:#### Page 25 of 1 ####

Soszy´nski et al. (2018). A significant reduction in dispersion is evident in the case of the optical PWR 8 LMC T2Cs K when compared to optical PLRs. The Ks-band PLRs from Matsunaga et al. (2006), Bhardwaj et al. (2017a), and Braga et al. (2018) are: H

12 MK = 1.10 2.41 log(P) (σ = 0.14), J s GGC − − K = 17.10 2.23 log(P) (σ = 0.18), s LMC − K = 13.44 2.23 log(P) (σ = 0.28). (21) s BLG − W V,I The slopes of the K -band PLRs are very similar

Magnitudes s 16 between the LMC and bulge short-period B Her + W Vir T2Cs. In Ks-band, Bhardwaj et al. (2017b) did not find a significant deviation in the slope of PLRs for RV I Tau from the B Her + W Vir sample but their photome- try for BL Her showed evidence of crowding effects as their target fields were in the central bar of the LMC. 20 At present, T2C PLRs are mainly calibrated with zero-point anchored to the LMC thanks to a very pre- 0 1 2 cise 1% late-type eclipsing binary distance. The HST log(P) parallaxes are available for only two T2Cs (κ Pav and VY Pyx) and there are two T2Cs with Baade-Wesselink Figure 12. Multiband PLRs and optical PWR for T2Cs in distance (Feast et al., 2008). Therefore, a robust Galac- the LMC. The shaded regions represent three subclasses of tic calibration is still lacking but is expected to be de- T2Cs. The dashed lines represent a linear regression fitted to livered with increasingly accurate astrometric data from BL Her and W Vir subclasses. The magnitudes in different the Gaia mission. bands are offset by some arbitrary amount for visualization Using T2Cs in the Galactic bulge from the VISTA purposes. VVV survey, Bhardwaj et al. (2017c) derived PLRs in JHKs-bands and estimated a robust distance to the Galactic center. T2Cs are particularly interesting in the extremely crowded regions like the Galactic bulge ing PLR fits exhibit large dispersion. This suggests that because their multiband NIR PLR and PWRs can be the contribution to the intrinsic scatter in T2C PLRs at used to constrain the individual distances and extinc- optical bands may have some dependence on metallic- tion simultaneously without accounting for the metal- ity and stellar environment. The presence of peculiar licity effects. For example, the distance distribution de- W Vir stars that have distinct light curve shapes also rived using PLRs without accounting metallicity effects contributes to the scatter in the PLRs as they are sys- is much broader for RR Lyrae in the bulge than T2Cs tematically brighter than W Vir stars. Soszy´nski et al. (Bhardwaj et al., 2017c) implying a better precision for (2017b) found that a significant fraction of W Vir are individual T2C distances. Recently, Braga et al. (2018) in the eclipsing binary systems and thus should be ex- extended this work with a larger sample of T2Cs in cluded from the PLR fits to obtain a better distance esti- the bulge and derived individual distances to trace the mates using T2Cs. In Figure 12, the PLRsin NIR bands structure and kinematics of old stellar populations (see do not show a significant deviation in the slope between also, D´ek´any et al., 2019). BL Her and W Vir subclasses. However, pW Vir and From theoretical point of view, Di Criscienzo et al. RV Tau are systematically brighter than the PLRs fol- (2007) derived PLR and PWRs for T2Cs and esti- lowed by short-period T2Cs. The T2C PLRs and PWRs mated distances to several GGCs that were found to for a combined sample of BL Her and W Vir stars in be consistent with RR Lyrae based estimates. They different stellar systems are provided here. also predicted that the slope of the overall PLRs for T2Cs is less steep than that of classical Cepheids, W = 17.32 2.49 log(P) (σ = 0.12), which is also seen in the empirical PLRs. The fact V,ILMC − that T2Cs follow similar PLRs as that of RR Lyrae W = 17.59 2.54 log(P) (σ = 0.38), (20) V,ISMC − in NIR (Matsunaga et al., 2006; Feast et al., 2012; where the optical photometry is taken from Bhardwaj et al., 2017b) suggests a continuous transi- #### Page 26 of 1 J. Astrophys. Astr. (0000) 000:#### tion between evolved RR Lyrae and BL Her evolu- properties. The evolutionary time scales of T2Cs are tionary and pulsational properties. Recent findings roughly two orders of magnitude faster than RR Lyrae have confirmed that BL Her and W Vir show similar (Marconi et al., 2015). Unlike classical Cepheids, the PLRs provided pW Vir are excluded from the sample optical PLRs of T2Cs are non-linear and lack the pre- (Ripepi et al., 2015; Bhardwaj et al., 2017b,c). How- cision to be useful distance indicators. T2C observa- ever, RV Tau are found to be systematically brighter tions in NIR, where their PLRs do not show significant than the PLRs followed by BL Her and W Vir stars. RV metallicity dependence (see Sec 6.2), are increasing as Tau are post-AGB stars and they may have circumstel- there are more NIR variability surveys. They are used lar envelopes which can make them fainter. However, both as population tracers and distance indicators com- this can lead to significant variation in luminosities in plementing RR Lyrae variables. T2Cs are brighter than different pulsation cycles and subsequently contribute RR Lyrae and therefore can extend the use of popu- to the scatter in the PLRs. lation II standard candles to galaxies beyond 2 Mpc where the application of RR Lyrae is presently lim- 6.2.1 Metallicity effects: The theoretical and ob- ited (Da Costa et al., 2010). The light curves of T2Cs servational investigations of T2Cs suggest mini- are not as distinct as classical Cepheids and RR Lyrae, mal or no dependence of metal-abundances on and a significant overlap with classical Cepheids can NIR PLRs unlike RR Lyrae. The metallicity ef- be seen on the Fourier plane and color-magnitude dia- fects on NIR PLRs of T2Cs are at the level grams. of 0.05 mag/dex according to the theoreti- Fig. 13 displays calibrated PLR in Ks-band for clas- cal predictions∼ (Bono, Caputo & Santolamazza, 1997; sical pulsating stars in the LMC. Classical Cepheids are Di Criscienzo et al., 2007; Marconi & Di Criscienzo, systematically 1.5-3 magnitude brighter than T2Cs 2007). Empirically, Matsunaga et al. (2006) found at a fixed period∼ but the exact difference is period- negligible effect of metallicity dependence on NIR dependent. Depending on the period of T2Cs, these are PLRs for T2Cs in the GGCs. Bhardwaj et al. (2017c) up to 8 mag brighter than population II RR Lyrae. The showed that the slope of K-band PLR of T2C is sta- BL Her and W Vir subclasses of T2Cs follow a linear tistically similar between GGCs, bulge, LMC and for PLRs. Extending their PLR to shorter periods (<1 day) the Milky Way T2Cs having good parallax measure- clearly suggests that RRab are also located on this re- ments. Spectroscopic measurements for T2Cs are very lation while the overtone RRc seem to be brighter than limited (for example, Maas, Giridhar & Lambert, 2007; the PLR of T2Cs. The RV Tau are typically not in- Lemasle et al., 2015; Kovtyukh et al., 2018, and refer- cluded in the PLRs fits for T2Cs. The calibration of ences therein) to investigate metallicity effects but these PLRs for classical Cepheids, RR Lyrae and T2Cs in the stars are known to cover a range of metallicities simi- LMC is based on the eclipsing binary distance to the lar to that of RRLs. Without accounting metallicity ef- LMC (Pietrzy´nski et al., 2019). fects, Ripepi et al. (2015); Bhardwaj et al. (2017b) de- termined a distance to the LMC based on empirical relations that is in excellent agreement with classical 7. Summary and Future Prospects Cepheid and RR Lyrae based estimates. Apart from the Magellanic Clouds I discussed observational and theoretical pulsational (Soszynski et al., 2008), T2Cs have also been properties of classical Cepheid, RR Lyrae and T2Cs, discovered in several other extragalactic stel- and their application to extragalactic distance mea- lar systems, for example, IC1613, M31, M33 surements. The first two sections presented a histori- (Majaess, Turner & Lane, 2009), and in dwarf cal overview and explained the evolutionary and pul- spheroidal galaxy Fornax (Bersier & Wood, 2002). sational scenario related to these classical pulsating Given that T2Cs are brighter than RR Lyrae, BL Her stars. The section that describes the light curve prop- and W Vir can be used to estimate distances to the erties provides an overview of their identification and galaxies beyond the Local Group up to ( 10 Mpc). classification as well as emphasises how their multi- However, long-term time-domain surveys are∼ critical to wavelength observations can constrain the stellar evo- identify and classify the T2Cs because of the complex lution and pulsation models. The last three sections light variations and a broad period range. are focussed on the use of classical pulsating stars for cosmic distance scale delineating both population I and 6.3 Comparison with classical Cepheids and RR Lyrae population II distance indicators. These standard can- dles have a long history dating back to more than a cen- T2Cs are not as abundant as classical Cepheids and tury and their theoretical and empirical investigations RR Lyrae due to their short evolutionary timescales, have persistently played significant roles in our under- which limits a detailed investigation of their pulsation J. Astrophys. Astr. (0000)000:#### Page 27 of 1 ####

-8 Cep FU BL Her Cep FO W Vir RRab RV Tau RRc s

K -4

0

-0.5 0.0 0.5 1.0 1.5 2.0 log(P)

Figure 13. The Ks-band PLR for classical Cepheids, RR Lyrae and T2Cs in the LMC calibrated with 1% precise late-type eclipsing binary distance from Pietrzy´nski et al. (2019). The dashed lines represent best-fit linear regression over fundamental and first-overtone mode classical Cepheids, and BL Her + W Vir sample of T2Cs.

standing of the stellar evolution, Galactic structure and release are presented in this manuscript, it became ap- the Universe. parent that the current data release suffers from a sig- A discussion on primary calibrations for classical nificant zero-point offset in parallaxes for both Cepheid Cepheids, RR Lyrae and T2Cs reveals two major is- and RR Lyrae (for example, Muraveva et al., 2018a; sues that are yet to be addressed properly. First, the Riess et al., 2018b). Regardless, the plethora of Gaia lack of robust geometric distances that limits the pre- astrometric data, spectro-photometry, variability and cision of calibrated PLRs of these standard candles in spectroscopy of bright sources will lead to potential our own Galaxy. The HST parallaxes are available for a breakthroughs in the studies of classical pulsating vari- small sample of 16 classical Cepheids (Benedict et al., able stars. 2007; Riess et al., 2014, 2018a), 5 RR Lyrae and 2 The second outstanding question is related to the T2Cs (Benedict et al., 2011). With such small statis- impact of composition, metallicity and helium effects tics even the precise determination of slope and zero- in particular, age and evolutionary effects on the pri- point of PLRs is not possible let alone quantifica- mary calibration of the classical pulsating stars. In tion of other systematics, for example, due to age or the course of writing this manuscript, it became evi- metallicity. Therefore, either the theoretical calibra- dent that the metallicity effects on both the theoretical tions are adopted or these standard candles in the LMC and empirical PLRs are not well constrained despite serve as primary calibrators. In case of the former, decades of efforts. The high-resolution spectroscopic theoretical predictions suffer from the lack of obser- observations for Cepheids and RR Lyrae are very lim- vational constraints while in the case of latter, lack ited and will not be available even with Gaia data. of high precision abundances for these stellar popu- Therefore, complementary large scale ongoing spectro- lations (Mancino et al., 2020) in the LMC precludes scopic surveys (e.g. APOGEE (Majewski et al., 2017), quantification of other systematic uncertainties. How- LAMOST (Zhao et al., 2012)) and future facilities (e.g. ever, Gaia mission is already providing unprecedently 4MOST, de Jong, 2019) will provide stellar parame- precise astrometry for stellar populations in the so- ters for a statistically significant sample of pulsating lar neighbourhood (Lindegren et al., 2018). The fi- stars. In case of the highly extincted and crowded extra- nal Gaia parallaxes are predicted to have 10% pre- galactic systems, a deeper insight in our understanding cise parallaxes at a distance of 10 kpc and < 2% of the physics and chemistry of the classical pulsating within a few kpc distance. Therefore, a robust cali- stars will come with higher sensitivity and resolution of bration of nearby Cepheid and RR Lyrae populations JWST in space and 30-m class ground-based extremely will eventually be derived with a percent-level preci- large telescopes. sion. While some of the results from Gaia second data High-precision space-based photometry has #### Page 28 of 1 J. Astrophys. Astr. (0000) 000:#### revealed interesting additional non-radial modes, (ground versus space-based), possible contributions of period-doubling and amplitude/phase modulations additional parameters to the intrinsic dispersion in the in classical pulsating stars (Kolenberg et al., 2010; PLRs. With predominantly infrared observational facil- Derekas et al., 2017; Moln´ar, 2018). The photometric ities in the future, classical pulsating variables will en- revolution with ongoing missions such as Gaia and able potential scientific discoveries related to the struc- TESS, and future PLATO mission will continue to ture of the Milky Way to the evolution of stars and our explore these phenomena in pulsating variables. The Universe. classical pulsators are less explored at UV and X-ray wavelengths where new insights can be gained into evolution and pulsation, and heating and dynamics of Acknowledgements their atmospheres (Engle, 2015; Neilson et al., 2016). At UV wavelengths, the large amplitudes are particu- I thank the editorial board of the Journal of Astro- larly interesting not only for the identification but also physics and Astronomy for inviting me to write this to provide constraints for the pulsation models, for review article and Marcio Catelan, Marina Rejkuba, example, by simultaneous model-fitting of multiband Wolfgang Gieren, H. P. Singh, Noriyuki Matsunaga, light curves. While the focus of this review is on Richard I. Anderson and Shashi Kanbur for useful com- distance measurements, classical pulsating stars are ments and suggestions. I also thank the anonymous ref- also used extensively as stellar population tracers for eree for the quick and constructive report that helped extinction, metallicity, and morphology of their host improve the manuscript. AB acknowledges research galaxies. For example, minimum light color of RR grant #11850410434 awarded by the National Natu- Lyrae is an excellent tool for reddening diagnostics ral Science Foundation of China through the Research (Sturch, 1966; Ngeow et al., 2017; Saha et al., 2019). Fund for International Young Scientists, and a China Classical Cepheid and RR Lyrae (and T2Cs) have Post-doctoral General Grant, and the Gruber fellowship been used to trace the spatial distribution and kine- 2020 grant sponsored by The Gruber Foundation and matics of young (metal-rich) and old (metal-poor) the International Astronomical Union. This research stellar populations in the Galaxy and the Magel- was supported by the Munich Institute for Astro- and lanic Clouds (see Subramanian & Subramaniam, Particle Physics (MIAPP) of the DFG cluster of excel- 2012; D´ek´any et al., 2013; Deb & Singh, lence “Origin and Structure of the Universe”. 2014; Subramanian & Subramaniam, 2015; Pietrukowicz et al., 2015; References Jacyszyn-Dobrzeniecka et al., 2016, 2017; Ripepi et al., 2017; Muraveva et al., 2018b; Alcock C. et al., 1998, AJ, 115, 1921 Skowron et al., 2019, and reference therein for Anderson R. I., 2016, MNRAS, 463, 1707 more details). Improved absolute calibrations of Anderson R. I., 2019, A&A, 631, A165 classical pulsating stars will enable precise individual Anderson R. I., Ekstr¨om S., Georgy C., Meynet G., distance measurements allowing new insights into the Mowlavi N., Eyer L., 2014, A&A, 564, A100 structure and kinematics of their underlying stellar Anderson R. I., Riess A. 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